mathematical modeling, analysis, and simulation of...
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Research ArticleMathematical Modeling Analysis and Simulation of TumorDynamics with Drug Interventions
Pranav Unni 1 and Padmanabhan Seshaiyer 2
1American International School Chennai Chennai Tamilnadu India2George Mason University Fairfax Virginia USA
Correspondence should be addressed to Padmanabhan Seshaiyer pseshaiygmuedu
Received 25 April 2019 Revised 7 August 2019 Accepted 5 September 2019 Published 8 October 2019
Guest Editor Madjid Soltani
Copyright copy 2019 Pranav Unni and Padmanabhan Seshaiyer )is is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited
Over the last few decades there have been significant developments in theoretical experimental and clinical approaches tounderstand the dynamics of cancer cells and their interactions with the immune system )ese have led to the development ofimportant methods for cancer therapy including virotherapy immunotherapy chemotherapy targeted drug therapy and manyothers Along with this there have also been some developments on analytical and computational models to help provide insightsinto clinical observations )is work develops a new mathematical model that combines important interactions between tumorcells and cells in the immune systems including natural killer cells dendritic cells and cytotoxic CD8+ Tcells combined with drugdelivery to these cell sites )ese interactions are described via a system of ordinary differential equations that are solvednumerically A stability analysis of this model is also performed to determine conditions for tumor-free equilibrium to be stableWe also study the influence of proliferation rates and drug interventions in the dynamics of all the cells involved Anothercontribution is the development of a novel parameter estimation methodology to determine optimal parameters in the model thatcan reproduce a given dataset Our results seem to suggest that the model employed is a robust candidate for studying thedynamics of tumor cells and it helps to provide the dynamic interactions between the tumor cells immune system and drug-response systems
1 Introduction
Cancer is one of the leading causes of death in the worldtoday By 2030 over 13 million are estimated to harbor someform of the disease While there have been many de-velopments in cancer therapies including surgery chemo-therapy immunotherapy and radiotherapy there is still a lotthat is unknown about the dynamics of how cancer cells arecreated propagated and destroyed
Over the past few decades there have been several ex-perimental approaches and interventions developed thathave helped us to understand the dynamics of tumor growthand its interactions with the immune system [1 2] )is hasalso helped to inform how specific interventions such asimmunotherapy can help strengthen our own ability to fightcancer by improving the effectiveness of the immune system
[3ndash5] While these developments have helped enhance ourunderstanding about cancer dynamics there are still severalchallenges in these experimental approaches to fully un-derstand the interactions with the immune system
In the last two decades there have also been severalexperimental advances in developing interventional thera-pies for cancer such as immunotherapy virotherapy tar-geted drug therapies and chemotherapy Along with theseexperimental developments there have been some advancesin scientific and engineering solutions to capture the dy-namics of cancer One of the promising approaches includesmathematical modeling [6 7] which involves identifying thecells that play a role in cancer propagation interactionsbetween these bodies and description of the dynamics ofthis interaction that has helped estimate parameters per-form stability analysis and predict tumor dynamics [8ndash13]
HindawiComputational and Mathematical Methods in MedicineVolume 2019 Article ID 4079298 13 pageshttpsdoiorg10115520194079298
)ese models have been able to demonstrate the importanceof the presence of immune components for explainingclinically observed phenomena such as tumor dormancy[14] tumor size oscillations and regressions [15ndash18] non-spatial models of tumor and immune system interactions[19 20] and tumor growth coupled with immunotherapy[8 9 12 13 21 22]
)ese mathematical models are often coupled systemof governing differential equations that describe the dy-namics of each of the interacting component cells Spe-cifically the interactions between tumor growth and theimmune system are often described using a system ofcoupled differential equations with prescribed initialconditions )ese equations include nonlinear in-teractions and do not often admit an exact solution andtherefore require computational methods to solve themWhile these mathematical models have provided usefulinformation regarding the importance of the immunesystem in controlling tumor growth there is still a greatneed to continue to enhance existing models to in-corporate new clinical developments and biological dis-coveries For example there have been studies suggestingthe effectiveness of chemotherapy with immunotherapyand vaccine therapies [1 13 23] )e focus of this paper isto enhance existing models of tumor growth that in-corporate tumor dynamics in conjunction with the im-mune system response and also study the effect ofadditional interventions including antitumor vaccinationand immunotherapies along with chemotherapy
Of the many clinical approaches that are tested forcancer therapy one of the popular approaches includesdrug therapy to the tumor microenvironment To un-derstand the impact of the drugs delivered to the tumor cellsite it is important to include the effect of these drugs intothe models as well Towards this end we develop amathematical model that will combine essential in-teractions between growing tumor cells and cells of theinnate and specific immune system coupled with modelsfor drug delivery to these cell sites Our goal is to use thesemodels developed to study the effectiveness of anticancerdrugs to reduce tumor growth
)e growth of tumors has also been attributed to thedynamics of the cellular immune system within the humanhost Two principal components of this immune systeminclude the natural killer cells and cytotoxic CD8+ T cellswhich are known to kill tumor cells Besides these otherimportant antigen-presenting cells include the dendriticcells that help stimulate recruit and activate the immunesystem While research has been growing to discover po-tential mechanisms to describe immune system interactionswith growing tumors there is enough evidence that thedynamics of natural kills cells cytotoxic CD8+ T-cells anddendritic cells influence tumor dynamics )e model pre-sented in this work will account for the influence of these aswell
Over the years there has been a lot of development inmathematical modeling of cancer However the mecha-nisms that are involved in the interactions of tumor cellswith the immune system are still not clear )is paper
attempts to make a new contribution in this direction bydeveloping a coupled mathematical model that incorporatestumor dynamics and interactions between the dendriticcells natural killer cells and CD8+ T cells Additionally themodel incorporates and studies the influence of various drugtherapies including immunotherapy and chemotherapyFinally a new parameter estimation technique is proposedthat helps to estimate parameters optimally for a givenextrapolated dataset
2 Models and Background
In this work we will consider a model that consists of fourmain cell populations including tumor cells (T(t)) naturalkiller cells (N(t)) dendritic cells (D(t)) and cytotoxicCD8+ Tcells denoted by (L(t)) )e dynamics of these cellswill include interactions between each other as well asdynamics generated by interaction with chemotherapy aswell immunotherapy drug concentrations in the bloodstream
For developing the model for each of the cell pop-ulations a standard approach is to begin with applyingconservation of mass with diffusion and activation )iswould often yield the following equation for the dynamics ofthe various types of cells
z[middot]
zt+ nabla middot ( u
rarr[middot]) minus δDnabla
2[middot] f(middot) minus g(middot) minus K[middot]z(M)[middot]
(1)
Here the functions f(middot) and g(middot) will be based onproliferation rates competition terms and inhibition termsbased on the respective roles of each type of cell
Also note that in all the models we will consider theeffect of a chemotherapy drug (dynamics described later) killterm through K[middot]z(M)[middot] )e term z(M) 1 minus eminus M is usedto denote the fact that chemotherapeutic drugs (for exampledoxorubicin) are only effective during certain phases of thecell cycle and pharmacokinetics )e values of the kill pa-rameters K[middot] for the four cell population considered here arebased on their ability to disrupt the process of division andgrowth [24 25] Note that if K[middot] 0 the equation is notimpacted by the drug kill term
While equation (1) includes both the diffusion term andthe advection term due to blood velocity u
rarr we will consideronly the temporal dynamics in this work and hence theassociated ordinary differential equation
d[middot]
dt f(middot) minus g(middot) minus K[middot]z(M)[middot] (2)
21 Modeling Tumor Cells We begin with modeling tumorcells T which are assumed to have a proliferation rate thatcan be modeled by a logistic growth law aT(1 minus bT) withparameters a and b denoting the per capita growth ratesand reciprocal carrying capacities of the tumor cells[8 9 26] Also it is known that the growth of the tumorcells is impacted by three different competitive interactionsincluding interactions between tumor cells and dendritic
2 Computational and Mathematical Methods in Medicine
cells interactions between tumor cells and natural killercells and interactions between tumor cells and CD8+ Tcells[26ndash28] Denoting the corresponding competition rates asj c1 k respectively introduces the competition term asminus (c1N + j D + kL)T )e dynamics of tumor cells can thenbe described by the following ordinary differentialequationdT
dt aT(1 minus bT) minus c1N + j D + kL( 1113857T minus KTz(M)T (3)
22 Modeling Natural Killer Cells To model natural killer(NK) cells we will assume that these cells have a constantsource s1 as well as a NK cell recruitment term that can berepresented through a modified MichaelisndashMenten term(commonly used to govern cell-cell interactions)
g1 middotT2
h1 + T2 middot N (4)
where g1 denotes the maximum NK cell recruitment rate bytumor cells and h1 denotes the steepness coefficient of theNK cell recruitment curve [8]
Next the growth of NK cells will be impacted by twodifferent interactions namely the interaction between NKcells and tumor cells [29] and the interaction between NKcells and dendritic cells [30ndash33] We also introduce pa-rameters c2 d1 to be the rates of killing (due to tumor cells)and proliferation (due to dendritic cells) of NK cells re-spectively )e governing differential equation for the dy-namics of NK cells then can be described asdN
dt s1 +
g1NT2
h1 + T2 minus c2T minus d1D( 1113857N minus KNz(M)N minus eN
(5)
Note that we have also included a natural death of NKcells through minus eN
23 Modeling Dendritic Cells Dendritic cells play an im-portant role in the immune system response and incontrolling tumor growth Also known as antigen-pre-senting cells they update and present antigens to CD8+
T cells Some of the earlier models [8 13] in the literaturehave not incorporated the dynamics of dendritic cells indirectly suppressing tumor growth stimulating restingNK cells and impacting the dynamics of CD8+ T cells)ere is however experimental evidence that dendriticcells play an important role in modeling tumor immu-notherapy [28]
To study the dynamics of dendritic cells we will assumes2 to be a constant source of dendritic cells d2 to be the rateat which NK cells kill dendritic cells d3 to be a proliferationrate of dendritic cells due to tumor cells f1 to be the ratecorresponding to the interaction of dendritic cells withCD8+ T cells and g to be the natural death rate of dendriticcells We then have
dD
dt s2 minus f1L + d2N minus d3T( 1113857D minus KDz(M)D minus gD (6)
24 Modeling Cytotoxic CD8+ T Cells Among many factorsthat impact the growth of tumor cells it is well known thatCD8+ T cells are an important component of the immunesystem that kills tumor cells It has been seen that activetumor-specific CD8+ T cells are only present in largenumbers when tumor cells are present [16 34] and aftersome interactions with tumor cells they become inactive[29] It has been observed that mature CD8+ T cells canremove dendritic cells [35 36]
In our model to describe the dynamics of the CD8+
T-cells we will consider f2 to be the rate of interactionbetween dendritic cells and tumor cells to activate CD8+
T cells minus hLT denotes the form of competition betweenCD8+ T cells and tumor cells and minus iL denotes the naturaldeath rate of CD8+ T cells
It has also been seen that CD8+ T cells may be recruitedby the debris from tumor cells lysed by NK cells [37] Toinclude this effect we will incorporate in our model a re-cruitment term that is proportional to the number of cellskilled which is denoted as r1NT We will also need anadditional term that helps describe the regulation andsuppression of CD8+ T-cell activity when there are highlevels of activated CD8+ T cells without responsiveness tocytokines in the system [38 39] )is term is denoted byuNL2 We also include CD8+ T activation by IL-2 immu-notherapy which is in the form of a drug that influences theimmune systemrsquos efficacy and described via a MichaelisndashMenten interaction term [16]
pILI
gI + I (7)
Here I refers to the immunotherapy drug concentrationin the bloodstream )e model for the dynamics for theCD8+ T cell growth population becomes
dL
dt f2DT minus hLT minus uNL
2+ r1NT +
pILI
gI + Iminus KLz(M)L minus iL
(8)
25 Modeling Drug and Vaccine Interventions In this studywe incorporate a variety of external intervention treatmentoptions including tumor-infiltrating lymphocyte (TIL) in-jections as well as chemotherapy and immunotherapy drugsTIL drug intervention may be thought of as an immuno-therapy approach in which the CD8+ T-cells are promotedthrough antigen-specific cytolytic immune cells We do thisby adding the term vL vL(t) in equation (8) and we havedL
dt f2DT minus hLT minus uNL
2+ r1NT +
pILI
gI + Iminus KLz(M)L minus iL + vL
(9)
Computational and Mathematical Methods in Medicine 3
To include the chemotherapy and immunotherapydrugs we describe the dynamics of the respective concen-trations in the blood stream as follows
dMdt
vM(t) minus d4M
dIdt vI(t) minus d5I
(10)
e drug intervention terms in these equations reectthe amount of chemotherapy and immunotherapy druggiven over time Note that we assume that the chemotherapyand immunotherapy drugs will be eliminated from the bodyover time at a rate proportional to its concentration andthese are given by d4M and d5I respectively
26 Overall Model From Sections 21ndash25 we have thefollowing overall model
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T
_N s1 +g1NT
2
h1 + T2 minus c2T minus d1D( )N minus KNz(M)N minus eN
_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD
_L f2DT minus hLT minus uNL2 + r1NT +
pILI
gI + I
minus KLz(M)L minus iL + vL(t)
_M vM(t) minus d4M
_I vI(t) minus d5I(11)
Figure 1 illustrates the network of the dynamics forsystem (11) Sharp arrows represent reproduction or acti-vation and blocked arrows represent inhibition or killinge blocked arrow in red represents the nonlinear in-teraction Note that the suppressing eects of the drug arenot shown explicitly in the gure but included in the model
3 Stability Analysis
In this section we employ mathematical analysis to identifyconditions that can help eliminate tumor cells Also we willdetermine conditions for when tumor-free equilibrium isunstable and the tumor grows without bound
We now consider the system of (11) in the absence oftreatment When we eliminate chemotherapy and immu-notherapy the system reduces to a four-population systemof ODEs Let E(Tlowast Nlowast Dlowast Llowast) be an equilibrium point ofthe system described by the system without drugintervention
At an equilibrium point we havedTdtdNdt
dDdt
dLdt 0 (12)
Since we assume there is constant recruitment throughsource terms s1 and s2 both not equal to zero there is notrivial equilibrium which implies
E Tlowast Nlowast Dlowast Llowast( )ne (0 0 0 0) (13)
For tumor-free equilibrium at an equilibrium point wehave dNdt 0 which yields
s1 + d1DlowastNlowast minus eNlowast 0 (14)
which yields
Nlowast s1
e minus d1Dlowast (15)
Similarly setting dDdt 0 at an equilibrium pointyields
s2 minus d2NlowastDlowast minus gDlowast 0 (16)
Substituting (15) in (16) we get
gd1Dlowast2 minus s2d1 + d2s1 + eg( )Dlowast + es2 0 (17)
Solving the quadratic equation yields
Dlowast12 d1s2 + d2s1 + eg( ) plusmn
d1s2 + d2s1 + eg( )2 minus 4ges2
radic
2gd1
(18)
Hence we have 2 tumor-free equilibrium given byE1(0 Nlowast Dlowast1 0) and E2(0 Nlowast Dlowast2 0)
For these to have biological meaning we need
e minus d1Dlowast gt 0 (19)
d1s2 + d2s1 + egge 2ges2
radic (20)
ese conditions suggest critical values for the death rateand the source term for NK cells to be
e d1Dlowast
s1 2 ges2
radic minus d1s2 + eg( )d2
(21)
in order for the tumor-free equilibrium point to be positivefor biological signicance
NK
CD8+ T
T
Chemotherapy Immunotherapy
D
Figure 1 Network of the dynamics for system (11)
4 Computational and Mathematical Methods in Medicine
)e Jacobian matrix for linearization of system (11)without drug intervention is given by
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(22)
where
a11 a minus 2abTlowast
minus c1Nlowast
minus jDlowast
minus kLlowast
a12 minus c1Tlowast
a13 minus jTlowast
a14 minus kTlowast
a21 2g1N
lowasth1Tlowast
h1 + T( 11138572 minus c2N
lowast
a22 minus c2Tlowast
+ d1Dlowast
minus e
a23 d1Nlowast
a24 0
a31 d3Dlowast
a32 minus d2Dlowast
a33 minus f1Llowast
+ d2Nlowast
minus d3Tlowast
+ g( 1113857
a34 minus f1Dlowast
a41 f2Dlowast
minus hLlowast
+ r1Nlowast
a42 minus uLlowast2
+ r1Tlowast
a43 f2Tlowast
a44 minus hTlowast
+ 2uNlowastLlowast
+ i( 1113857
(23)
Evaluating these terms at the general tumor-free equi-librium point gives
A
a minus c1Nlowast minus jDlowast 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(24)
To solve for the eigenvalues λ we solve the equationdet(A minus λI) 0 or
det
a minus c1Nlowast minus jDlowast minus λ 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e minus λ d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus λ minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i minus λ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
0
(25)which yields
a minus c1Nlowast
minus jDlowast
minus λ( 1113857(minus i minus λ)det(B minus λI) 0 (26)
where matrix B is given by
B d1Dlowast minus e d1N
lowast
minus d2Dlowast minus d2N
lowast + g( 11138571113890 1113891 (27)
It may be noted that trace and determinant for matrix Bcan be computed to be
tr(B) d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 1113857 (28)
det(B) d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 (29)
Solving (26) we compute the eigenvalues to be
λ1 a minus c1Nlowast
minus jDlowast
λ2 minus i(30)
and λ3 and λ4 to be the roots of the equation
λ2 minus λ d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 11138571113858 1113859 + d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 0(31)
From (28) and (29) this can be rewritten as
λ2 minus tr(B)λ + det(B) 0 (32)
Using (19) in equations (28) and (29) we can concludethat
tr(B)lt 0
det(B)gt 0(33)
)is implies that the eigenvalues λ3 and λ4 that are theroots of equation (32) have negative real parts
In order for the tumor-free equilibrium to be stable werequire λ1 lt 0 which implies that if the tumor growth rate ais lesser than the critical value given by c1N
lowast + JDlowast then thetumor population can be eliminated
4 Computational Experiments
In this section we will consider the system of (11) which willbe solved via the RungendashKutta methods )e parametervalues their units and their estimated value are itemizednext which we use in our computations
(i) a tumor growth rate estimated as431lowast10minus 1 dayminus 1 [9]
(ii) b bminus 1 tumor-carrying capacity estimated as217lowast10minus 8 cellsminus 1 [9]
Computational and Mathematical Methods in Medicine 5
(iii) c1 NK cell tumor cell kill rate estimated as35lowast 10minus 6 cellsminus 1 [12]
(iv) c2 NK cell inactivation rate by tumor cells esti-mated as 10lowast 10minus 7 cellsminus 1 dayminus 1 [9]
(v) d1 rate of dendritic cell priming NK cells esti-mated as 10lowast 10minus 6 cellsminus 1 [12]
(vi) d2 NK cell dendritic cell kill rate estimated as40lowast 10minus 6 cellsminus 1 [12]
(vii) d3 rate of tumor cells priming dendritic cellsestimated as 10lowast 10minus 4 Estimate
(viii) e death rate of NK cell estimated as412lowast 10minus 2 dayminus 1 [12]
(ix) f1 CD8+ T cell dendritic cells kill rate estimated
as 10lowast 10minus 8 cellsminus 1 [12](x) f2 rate of dendritic cells priming CD8+ T cell
estimated as 001 cellsminus 1 [12](xi) g death rate of dendritic cells estimated as
24lowast 10minus 2 cellsminus 1 [12](xii) h CD8+ T inactivation rate by tumor cells esti-
mated as 342lowast 10minus 10 cellsminus 1 dayminus 1 [12](xiii) i death rate of CD8+ T cells estimated as
20lowast 10minus 2 dayminus 1 [9](xiv) j dendritic cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xv) k NK cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xvi) s1 source of NK cells estimated as 13lowast 104 cellsminus 1
[12](xvii) s2 source of dendritic cell estimated as
48lowast 102 cellsminus 1 [12](xviii) u regulatory function by Nk cells of CD8+ T cells
estimated as 180lowast 10minus 8 cellminus 2 dayminus 1 [9]
For the first computation we will assume there is noadditional recruitment terms for CD8+ T cells and NK cells(r1 0 g1 0 h1 0) removing some of the regulationsuppression and activation of CD8+ T cells(pI gI u 0) not including the influence of drug killterms (KT KN KD KL 0) no influence of drug andvaccine interventions (vL vM vI 0) along with thecorresponding death rates (d4 d5 0) We will also as-sume d3 0 which corresponds to the new term that hasbeen added to the model to indicate the growth of dendriticcells being impacted by tumor cells We will also assume forsimplicity and illustration purposes of a weak immunesystem that the dynamics start with 100 tumor cells with onenatural killer one dendritic and one CD8+ T cell Figure 2shows the dynamics of each of these cells )e tumor cellsinitially increase to a peak before a full immune clearancestarts
Next we consider the effect of one of the terms insystem (11) corresponding to the dynamics of dendriticcells )is is the term d3TD which includes the influence oftumor growth on the dynamics of dendritic cells that all
the previous studies have not considered Figure 3 illus-trates how not only the dendritic cells are impacted but theCD8+ Tcell dynamics also changes as the proliferation rated3 is doubled For the rest of the simulations we willinclude the effect of d3 and assume the value to be 1 times 10minus 4
Along with d3 we also wanted to study the influence ofthe source term s2 on the dynamics of tumor NK and CD8+
T cells Figure 4 illustrates this behavior When we increasethe source term of dendritic cells it increases NK cells andCD8+ Tcells We also note that these have an effect on tumorgrowth as both NK and CD8+ T cells can lyse tumor cells)is decrease is also seen in this figure )is suggests that anexternal source term of dendritic cells has the potential todecrease tumor growthWe also note that dendritic cells playan important role in recruiting CD8+ T cells earlier in thetumor growth phase
Next to study the effect of TIL drug intervention termonly for the CD8+ T-cell population as an immuno-therapy where the immune cell levels are boosted by theaddition of antigen-specific cytolytic immune cells weincrease the value of vL from 1 to 106 )e result is shownin Figure 5 and we notice that the effect of adding thedrug in small doses does not have a big impact on tumorgrowth
Next we consider the effect of the chemotherapy drugonly that is introduced through the term vM We set vM 1and study the influence of increasing KT in the dynamics oftumor cells Figure 6 illustrates how tumor cells can bereduced through this technique
We want to point out that we also performed the studyon the influence of immunotherapy drug intervention vI
but noticed that even with inclusion of a CD8+ Tactivation described via MichaelisndashMenten interaction givenby
pILI
gI + I (34)
there was negligible effect on the dynamics of all four cellsWe also noted that there was not much effect in the dy-namics of NK cells through the recruitment terms involvingg1 and r1
Next we turn our attention to the effect of the nonlinearterm introduced as an inactivation term which describes theregulation and suppression of CD8+ T-cell activity [38 39]Specifically Figure 7 shows the effect of the nonlinear termin system (11) for parameters u 0 and u 3 times 10minus 10 andthe dynamics show that there is a drastic drop in the numberof cytotoxic CD8+ Tcells while a small increase in tumor cellgrowth
In summary our computations seem to suggest that acombination of immunotherapy through TIL drug in-tervention vM along with chemotherapy through vL providesan optimal way to reduce tumor growth Taking this intoaccount and removing terms that had negligible effects onecan consider the following simplified system that capturesmost prominent features
6 Computational and Mathematical Methods in Medicine
0 50 100Time (days)
0
500
1000
1500
Tum
or ce
lls
(a)
0 50 100Time (days)
0
1
2
3
4
NK
cells
times105
(b)
0 50 100Time (days)
0
500
1000
1500
2000
Den
driti
c cel
ls
(c)
0 50 100Time (days)
0
5
10
15
CD8+ T
cells
times104
(d)
Figure 2 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells
0
500
1000
1500
Tum
or ce
lls
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(a)
0
1
2
3
4
NK
cells
times105
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(b)
Figure 3 Continued
Computational and Mathematical Methods in Medicine 7
0
1
2
3
Den
driti
c cel
ls
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
times104
0 50 100Time (days)
(c)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
0
5
10
15
CD8+ T
cells
times105
0 50 100Time (days)
(d)
Figure 3 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells as the proliferation rate d3 is doubled
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
s2 = 500s2 = 2000s2 = 5000
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
2
4
NK
cells
times105
s2 = 500s2 = 2000s2 = 5000
(b)
0 10 20 30 40 50 60 70 80 90 100Time (days)
s2 = 500s2 = 2000s2 = 5000
times105
0
5
10
15
CD8+ T
cells
(c)
Figure 4 Dynamics of (a) tumor (b) NK and (c) CD8+ T cells as function of source s2
8 Computational and Mathematical Methods in Medicine
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
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Submit your manuscripts atwwwhindawicom
)ese models have been able to demonstrate the importanceof the presence of immune components for explainingclinically observed phenomena such as tumor dormancy[14] tumor size oscillations and regressions [15ndash18] non-spatial models of tumor and immune system interactions[19 20] and tumor growth coupled with immunotherapy[8 9 12 13 21 22]
)ese mathematical models are often coupled systemof governing differential equations that describe the dy-namics of each of the interacting component cells Spe-cifically the interactions between tumor growth and theimmune system are often described using a system ofcoupled differential equations with prescribed initialconditions )ese equations include nonlinear in-teractions and do not often admit an exact solution andtherefore require computational methods to solve themWhile these mathematical models have provided usefulinformation regarding the importance of the immunesystem in controlling tumor growth there is still a greatneed to continue to enhance existing models to in-corporate new clinical developments and biological dis-coveries For example there have been studies suggestingthe effectiveness of chemotherapy with immunotherapyand vaccine therapies [1 13 23] )e focus of this paper isto enhance existing models of tumor growth that in-corporate tumor dynamics in conjunction with the im-mune system response and also study the effect ofadditional interventions including antitumor vaccinationand immunotherapies along with chemotherapy
Of the many clinical approaches that are tested forcancer therapy one of the popular approaches includesdrug therapy to the tumor microenvironment To un-derstand the impact of the drugs delivered to the tumor cellsite it is important to include the effect of these drugs intothe models as well Towards this end we develop amathematical model that will combine essential in-teractions between growing tumor cells and cells of theinnate and specific immune system coupled with modelsfor drug delivery to these cell sites Our goal is to use thesemodels developed to study the effectiveness of anticancerdrugs to reduce tumor growth
)e growth of tumors has also been attributed to thedynamics of the cellular immune system within the humanhost Two principal components of this immune systeminclude the natural killer cells and cytotoxic CD8+ T cellswhich are known to kill tumor cells Besides these otherimportant antigen-presenting cells include the dendriticcells that help stimulate recruit and activate the immunesystem While research has been growing to discover po-tential mechanisms to describe immune system interactionswith growing tumors there is enough evidence that thedynamics of natural kills cells cytotoxic CD8+ T-cells anddendritic cells influence tumor dynamics )e model pre-sented in this work will account for the influence of these aswell
Over the years there has been a lot of development inmathematical modeling of cancer However the mecha-nisms that are involved in the interactions of tumor cellswith the immune system are still not clear )is paper
attempts to make a new contribution in this direction bydeveloping a coupled mathematical model that incorporatestumor dynamics and interactions between the dendriticcells natural killer cells and CD8+ T cells Additionally themodel incorporates and studies the influence of various drugtherapies including immunotherapy and chemotherapyFinally a new parameter estimation technique is proposedthat helps to estimate parameters optimally for a givenextrapolated dataset
2 Models and Background
In this work we will consider a model that consists of fourmain cell populations including tumor cells (T(t)) naturalkiller cells (N(t)) dendritic cells (D(t)) and cytotoxicCD8+ Tcells denoted by (L(t)) )e dynamics of these cellswill include interactions between each other as well asdynamics generated by interaction with chemotherapy aswell immunotherapy drug concentrations in the bloodstream
For developing the model for each of the cell pop-ulations a standard approach is to begin with applyingconservation of mass with diffusion and activation )iswould often yield the following equation for the dynamics ofthe various types of cells
z[middot]
zt+ nabla middot ( u
rarr[middot]) minus δDnabla
2[middot] f(middot) minus g(middot) minus K[middot]z(M)[middot]
(1)
Here the functions f(middot) and g(middot) will be based onproliferation rates competition terms and inhibition termsbased on the respective roles of each type of cell
Also note that in all the models we will consider theeffect of a chemotherapy drug (dynamics described later) killterm through K[middot]z(M)[middot] )e term z(M) 1 minus eminus M is usedto denote the fact that chemotherapeutic drugs (for exampledoxorubicin) are only effective during certain phases of thecell cycle and pharmacokinetics )e values of the kill pa-rameters K[middot] for the four cell population considered here arebased on their ability to disrupt the process of division andgrowth [24 25] Note that if K[middot] 0 the equation is notimpacted by the drug kill term
While equation (1) includes both the diffusion term andthe advection term due to blood velocity u
rarr we will consideronly the temporal dynamics in this work and hence theassociated ordinary differential equation
d[middot]
dt f(middot) minus g(middot) minus K[middot]z(M)[middot] (2)
21 Modeling Tumor Cells We begin with modeling tumorcells T which are assumed to have a proliferation rate thatcan be modeled by a logistic growth law aT(1 minus bT) withparameters a and b denoting the per capita growth ratesand reciprocal carrying capacities of the tumor cells[8 9 26] Also it is known that the growth of the tumorcells is impacted by three different competitive interactionsincluding interactions between tumor cells and dendritic
2 Computational and Mathematical Methods in Medicine
cells interactions between tumor cells and natural killercells and interactions between tumor cells and CD8+ Tcells[26ndash28] Denoting the corresponding competition rates asj c1 k respectively introduces the competition term asminus (c1N + j D + kL)T )e dynamics of tumor cells can thenbe described by the following ordinary differentialequationdT
dt aT(1 minus bT) minus c1N + j D + kL( 1113857T minus KTz(M)T (3)
22 Modeling Natural Killer Cells To model natural killer(NK) cells we will assume that these cells have a constantsource s1 as well as a NK cell recruitment term that can berepresented through a modified MichaelisndashMenten term(commonly used to govern cell-cell interactions)
g1 middotT2
h1 + T2 middot N (4)
where g1 denotes the maximum NK cell recruitment rate bytumor cells and h1 denotes the steepness coefficient of theNK cell recruitment curve [8]
Next the growth of NK cells will be impacted by twodifferent interactions namely the interaction between NKcells and tumor cells [29] and the interaction between NKcells and dendritic cells [30ndash33] We also introduce pa-rameters c2 d1 to be the rates of killing (due to tumor cells)and proliferation (due to dendritic cells) of NK cells re-spectively )e governing differential equation for the dy-namics of NK cells then can be described asdN
dt s1 +
g1NT2
h1 + T2 minus c2T minus d1D( 1113857N minus KNz(M)N minus eN
(5)
Note that we have also included a natural death of NKcells through minus eN
23 Modeling Dendritic Cells Dendritic cells play an im-portant role in the immune system response and incontrolling tumor growth Also known as antigen-pre-senting cells they update and present antigens to CD8+
T cells Some of the earlier models [8 13] in the literaturehave not incorporated the dynamics of dendritic cells indirectly suppressing tumor growth stimulating restingNK cells and impacting the dynamics of CD8+ T cells)ere is however experimental evidence that dendriticcells play an important role in modeling tumor immu-notherapy [28]
To study the dynamics of dendritic cells we will assumes2 to be a constant source of dendritic cells d2 to be the rateat which NK cells kill dendritic cells d3 to be a proliferationrate of dendritic cells due to tumor cells f1 to be the ratecorresponding to the interaction of dendritic cells withCD8+ T cells and g to be the natural death rate of dendriticcells We then have
dD
dt s2 minus f1L + d2N minus d3T( 1113857D minus KDz(M)D minus gD (6)
24 Modeling Cytotoxic CD8+ T Cells Among many factorsthat impact the growth of tumor cells it is well known thatCD8+ T cells are an important component of the immunesystem that kills tumor cells It has been seen that activetumor-specific CD8+ T cells are only present in largenumbers when tumor cells are present [16 34] and aftersome interactions with tumor cells they become inactive[29] It has been observed that mature CD8+ T cells canremove dendritic cells [35 36]
In our model to describe the dynamics of the CD8+
T-cells we will consider f2 to be the rate of interactionbetween dendritic cells and tumor cells to activate CD8+
T cells minus hLT denotes the form of competition betweenCD8+ T cells and tumor cells and minus iL denotes the naturaldeath rate of CD8+ T cells
It has also been seen that CD8+ T cells may be recruitedby the debris from tumor cells lysed by NK cells [37] Toinclude this effect we will incorporate in our model a re-cruitment term that is proportional to the number of cellskilled which is denoted as r1NT We will also need anadditional term that helps describe the regulation andsuppression of CD8+ T-cell activity when there are highlevels of activated CD8+ T cells without responsiveness tocytokines in the system [38 39] )is term is denoted byuNL2 We also include CD8+ T activation by IL-2 immu-notherapy which is in the form of a drug that influences theimmune systemrsquos efficacy and described via a MichaelisndashMenten interaction term [16]
pILI
gI + I (7)
Here I refers to the immunotherapy drug concentrationin the bloodstream )e model for the dynamics for theCD8+ T cell growth population becomes
dL
dt f2DT minus hLT minus uNL
2+ r1NT +
pILI
gI + Iminus KLz(M)L minus iL
(8)
25 Modeling Drug and Vaccine Interventions In this studywe incorporate a variety of external intervention treatmentoptions including tumor-infiltrating lymphocyte (TIL) in-jections as well as chemotherapy and immunotherapy drugsTIL drug intervention may be thought of as an immuno-therapy approach in which the CD8+ T-cells are promotedthrough antigen-specific cytolytic immune cells We do thisby adding the term vL vL(t) in equation (8) and we havedL
dt f2DT minus hLT minus uNL
2+ r1NT +
pILI
gI + Iminus KLz(M)L minus iL + vL
(9)
Computational and Mathematical Methods in Medicine 3
To include the chemotherapy and immunotherapydrugs we describe the dynamics of the respective concen-trations in the blood stream as follows
dMdt
vM(t) minus d4M
dIdt vI(t) minus d5I
(10)
e drug intervention terms in these equations reectthe amount of chemotherapy and immunotherapy druggiven over time Note that we assume that the chemotherapyand immunotherapy drugs will be eliminated from the bodyover time at a rate proportional to its concentration andthese are given by d4M and d5I respectively
26 Overall Model From Sections 21ndash25 we have thefollowing overall model
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T
_N s1 +g1NT
2
h1 + T2 minus c2T minus d1D( )N minus KNz(M)N minus eN
_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD
_L f2DT minus hLT minus uNL2 + r1NT +
pILI
gI + I
minus KLz(M)L minus iL + vL(t)
_M vM(t) minus d4M
_I vI(t) minus d5I(11)
Figure 1 illustrates the network of the dynamics forsystem (11) Sharp arrows represent reproduction or acti-vation and blocked arrows represent inhibition or killinge blocked arrow in red represents the nonlinear in-teraction Note that the suppressing eects of the drug arenot shown explicitly in the gure but included in the model
3 Stability Analysis
In this section we employ mathematical analysis to identifyconditions that can help eliminate tumor cells Also we willdetermine conditions for when tumor-free equilibrium isunstable and the tumor grows without bound
We now consider the system of (11) in the absence oftreatment When we eliminate chemotherapy and immu-notherapy the system reduces to a four-population systemof ODEs Let E(Tlowast Nlowast Dlowast Llowast) be an equilibrium point ofthe system described by the system without drugintervention
At an equilibrium point we havedTdtdNdt
dDdt
dLdt 0 (12)
Since we assume there is constant recruitment throughsource terms s1 and s2 both not equal to zero there is notrivial equilibrium which implies
E Tlowast Nlowast Dlowast Llowast( )ne (0 0 0 0) (13)
For tumor-free equilibrium at an equilibrium point wehave dNdt 0 which yields
s1 + d1DlowastNlowast minus eNlowast 0 (14)
which yields
Nlowast s1
e minus d1Dlowast (15)
Similarly setting dDdt 0 at an equilibrium pointyields
s2 minus d2NlowastDlowast minus gDlowast 0 (16)
Substituting (15) in (16) we get
gd1Dlowast2 minus s2d1 + d2s1 + eg( )Dlowast + es2 0 (17)
Solving the quadratic equation yields
Dlowast12 d1s2 + d2s1 + eg( ) plusmn
d1s2 + d2s1 + eg( )2 minus 4ges2
radic
2gd1
(18)
Hence we have 2 tumor-free equilibrium given byE1(0 Nlowast Dlowast1 0) and E2(0 Nlowast Dlowast2 0)
For these to have biological meaning we need
e minus d1Dlowast gt 0 (19)
d1s2 + d2s1 + egge 2ges2
radic (20)
ese conditions suggest critical values for the death rateand the source term for NK cells to be
e d1Dlowast
s1 2 ges2
radic minus d1s2 + eg( )d2
(21)
in order for the tumor-free equilibrium point to be positivefor biological signicance
NK
CD8+ T
T
Chemotherapy Immunotherapy
D
Figure 1 Network of the dynamics for system (11)
4 Computational and Mathematical Methods in Medicine
)e Jacobian matrix for linearization of system (11)without drug intervention is given by
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(22)
where
a11 a minus 2abTlowast
minus c1Nlowast
minus jDlowast
minus kLlowast
a12 minus c1Tlowast
a13 minus jTlowast
a14 minus kTlowast
a21 2g1N
lowasth1Tlowast
h1 + T( 11138572 minus c2N
lowast
a22 minus c2Tlowast
+ d1Dlowast
minus e
a23 d1Nlowast
a24 0
a31 d3Dlowast
a32 minus d2Dlowast
a33 minus f1Llowast
+ d2Nlowast
minus d3Tlowast
+ g( 1113857
a34 minus f1Dlowast
a41 f2Dlowast
minus hLlowast
+ r1Nlowast
a42 minus uLlowast2
+ r1Tlowast
a43 f2Tlowast
a44 minus hTlowast
+ 2uNlowastLlowast
+ i( 1113857
(23)
Evaluating these terms at the general tumor-free equi-librium point gives
A
a minus c1Nlowast minus jDlowast 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(24)
To solve for the eigenvalues λ we solve the equationdet(A minus λI) 0 or
det
a minus c1Nlowast minus jDlowast minus λ 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e minus λ d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus λ minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i minus λ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
0
(25)which yields
a minus c1Nlowast
minus jDlowast
minus λ( 1113857(minus i minus λ)det(B minus λI) 0 (26)
where matrix B is given by
B d1Dlowast minus e d1N
lowast
minus d2Dlowast minus d2N
lowast + g( 11138571113890 1113891 (27)
It may be noted that trace and determinant for matrix Bcan be computed to be
tr(B) d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 1113857 (28)
det(B) d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 (29)
Solving (26) we compute the eigenvalues to be
λ1 a minus c1Nlowast
minus jDlowast
λ2 minus i(30)
and λ3 and λ4 to be the roots of the equation
λ2 minus λ d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 11138571113858 1113859 + d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 0(31)
From (28) and (29) this can be rewritten as
λ2 minus tr(B)λ + det(B) 0 (32)
Using (19) in equations (28) and (29) we can concludethat
tr(B)lt 0
det(B)gt 0(33)
)is implies that the eigenvalues λ3 and λ4 that are theroots of equation (32) have negative real parts
In order for the tumor-free equilibrium to be stable werequire λ1 lt 0 which implies that if the tumor growth rate ais lesser than the critical value given by c1N
lowast + JDlowast then thetumor population can be eliminated
4 Computational Experiments
In this section we will consider the system of (11) which willbe solved via the RungendashKutta methods )e parametervalues their units and their estimated value are itemizednext which we use in our computations
(i) a tumor growth rate estimated as431lowast10minus 1 dayminus 1 [9]
(ii) b bminus 1 tumor-carrying capacity estimated as217lowast10minus 8 cellsminus 1 [9]
Computational and Mathematical Methods in Medicine 5
(iii) c1 NK cell tumor cell kill rate estimated as35lowast 10minus 6 cellsminus 1 [12]
(iv) c2 NK cell inactivation rate by tumor cells esti-mated as 10lowast 10minus 7 cellsminus 1 dayminus 1 [9]
(v) d1 rate of dendritic cell priming NK cells esti-mated as 10lowast 10minus 6 cellsminus 1 [12]
(vi) d2 NK cell dendritic cell kill rate estimated as40lowast 10minus 6 cellsminus 1 [12]
(vii) d3 rate of tumor cells priming dendritic cellsestimated as 10lowast 10minus 4 Estimate
(viii) e death rate of NK cell estimated as412lowast 10minus 2 dayminus 1 [12]
(ix) f1 CD8+ T cell dendritic cells kill rate estimated
as 10lowast 10minus 8 cellsminus 1 [12](x) f2 rate of dendritic cells priming CD8+ T cell
estimated as 001 cellsminus 1 [12](xi) g death rate of dendritic cells estimated as
24lowast 10minus 2 cellsminus 1 [12](xii) h CD8+ T inactivation rate by tumor cells esti-
mated as 342lowast 10minus 10 cellsminus 1 dayminus 1 [12](xiii) i death rate of CD8+ T cells estimated as
20lowast 10minus 2 dayminus 1 [9](xiv) j dendritic cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xv) k NK cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xvi) s1 source of NK cells estimated as 13lowast 104 cellsminus 1
[12](xvii) s2 source of dendritic cell estimated as
48lowast 102 cellsminus 1 [12](xviii) u regulatory function by Nk cells of CD8+ T cells
estimated as 180lowast 10minus 8 cellminus 2 dayminus 1 [9]
For the first computation we will assume there is noadditional recruitment terms for CD8+ T cells and NK cells(r1 0 g1 0 h1 0) removing some of the regulationsuppression and activation of CD8+ T cells(pI gI u 0) not including the influence of drug killterms (KT KN KD KL 0) no influence of drug andvaccine interventions (vL vM vI 0) along with thecorresponding death rates (d4 d5 0) We will also as-sume d3 0 which corresponds to the new term that hasbeen added to the model to indicate the growth of dendriticcells being impacted by tumor cells We will also assume forsimplicity and illustration purposes of a weak immunesystem that the dynamics start with 100 tumor cells with onenatural killer one dendritic and one CD8+ T cell Figure 2shows the dynamics of each of these cells )e tumor cellsinitially increase to a peak before a full immune clearancestarts
Next we consider the effect of one of the terms insystem (11) corresponding to the dynamics of dendriticcells )is is the term d3TD which includes the influence oftumor growth on the dynamics of dendritic cells that all
the previous studies have not considered Figure 3 illus-trates how not only the dendritic cells are impacted but theCD8+ Tcell dynamics also changes as the proliferation rated3 is doubled For the rest of the simulations we willinclude the effect of d3 and assume the value to be 1 times 10minus 4
Along with d3 we also wanted to study the influence ofthe source term s2 on the dynamics of tumor NK and CD8+
T cells Figure 4 illustrates this behavior When we increasethe source term of dendritic cells it increases NK cells andCD8+ Tcells We also note that these have an effect on tumorgrowth as both NK and CD8+ T cells can lyse tumor cells)is decrease is also seen in this figure )is suggests that anexternal source term of dendritic cells has the potential todecrease tumor growthWe also note that dendritic cells playan important role in recruiting CD8+ T cells earlier in thetumor growth phase
Next to study the effect of TIL drug intervention termonly for the CD8+ T-cell population as an immuno-therapy where the immune cell levels are boosted by theaddition of antigen-specific cytolytic immune cells weincrease the value of vL from 1 to 106 )e result is shownin Figure 5 and we notice that the effect of adding thedrug in small doses does not have a big impact on tumorgrowth
Next we consider the effect of the chemotherapy drugonly that is introduced through the term vM We set vM 1and study the influence of increasing KT in the dynamics oftumor cells Figure 6 illustrates how tumor cells can bereduced through this technique
We want to point out that we also performed the studyon the influence of immunotherapy drug intervention vI
but noticed that even with inclusion of a CD8+ Tactivation described via MichaelisndashMenten interaction givenby
pILI
gI + I (34)
there was negligible effect on the dynamics of all four cellsWe also noted that there was not much effect in the dy-namics of NK cells through the recruitment terms involvingg1 and r1
Next we turn our attention to the effect of the nonlinearterm introduced as an inactivation term which describes theregulation and suppression of CD8+ T-cell activity [38 39]Specifically Figure 7 shows the effect of the nonlinear termin system (11) for parameters u 0 and u 3 times 10minus 10 andthe dynamics show that there is a drastic drop in the numberof cytotoxic CD8+ Tcells while a small increase in tumor cellgrowth
In summary our computations seem to suggest that acombination of immunotherapy through TIL drug in-tervention vM along with chemotherapy through vL providesan optimal way to reduce tumor growth Taking this intoaccount and removing terms that had negligible effects onecan consider the following simplified system that capturesmost prominent features
6 Computational and Mathematical Methods in Medicine
0 50 100Time (days)
0
500
1000
1500
Tum
or ce
lls
(a)
0 50 100Time (days)
0
1
2
3
4
NK
cells
times105
(b)
0 50 100Time (days)
0
500
1000
1500
2000
Den
driti
c cel
ls
(c)
0 50 100Time (days)
0
5
10
15
CD8+ T
cells
times104
(d)
Figure 2 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells
0
500
1000
1500
Tum
or ce
lls
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(a)
0
1
2
3
4
NK
cells
times105
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(b)
Figure 3 Continued
Computational and Mathematical Methods in Medicine 7
0
1
2
3
Den
driti
c cel
ls
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
times104
0 50 100Time (days)
(c)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
0
5
10
15
CD8+ T
cells
times105
0 50 100Time (days)
(d)
Figure 3 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells as the proliferation rate d3 is doubled
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
s2 = 500s2 = 2000s2 = 5000
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
2
4
NK
cells
times105
s2 = 500s2 = 2000s2 = 5000
(b)
0 10 20 30 40 50 60 70 80 90 100Time (days)
s2 = 500s2 = 2000s2 = 5000
times105
0
5
10
15
CD8+ T
cells
(c)
Figure 4 Dynamics of (a) tumor (b) NK and (c) CD8+ T cells as function of source s2
8 Computational and Mathematical Methods in Medicine
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
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Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
cells interactions between tumor cells and natural killercells and interactions between tumor cells and CD8+ Tcells[26ndash28] Denoting the corresponding competition rates asj c1 k respectively introduces the competition term asminus (c1N + j D + kL)T )e dynamics of tumor cells can thenbe described by the following ordinary differentialequationdT
dt aT(1 minus bT) minus c1N + j D + kL( 1113857T minus KTz(M)T (3)
22 Modeling Natural Killer Cells To model natural killer(NK) cells we will assume that these cells have a constantsource s1 as well as a NK cell recruitment term that can berepresented through a modified MichaelisndashMenten term(commonly used to govern cell-cell interactions)
g1 middotT2
h1 + T2 middot N (4)
where g1 denotes the maximum NK cell recruitment rate bytumor cells and h1 denotes the steepness coefficient of theNK cell recruitment curve [8]
Next the growth of NK cells will be impacted by twodifferent interactions namely the interaction between NKcells and tumor cells [29] and the interaction between NKcells and dendritic cells [30ndash33] We also introduce pa-rameters c2 d1 to be the rates of killing (due to tumor cells)and proliferation (due to dendritic cells) of NK cells re-spectively )e governing differential equation for the dy-namics of NK cells then can be described asdN
dt s1 +
g1NT2
h1 + T2 minus c2T minus d1D( 1113857N minus KNz(M)N minus eN
(5)
Note that we have also included a natural death of NKcells through minus eN
23 Modeling Dendritic Cells Dendritic cells play an im-portant role in the immune system response and incontrolling tumor growth Also known as antigen-pre-senting cells they update and present antigens to CD8+
T cells Some of the earlier models [8 13] in the literaturehave not incorporated the dynamics of dendritic cells indirectly suppressing tumor growth stimulating restingNK cells and impacting the dynamics of CD8+ T cells)ere is however experimental evidence that dendriticcells play an important role in modeling tumor immu-notherapy [28]
To study the dynamics of dendritic cells we will assumes2 to be a constant source of dendritic cells d2 to be the rateat which NK cells kill dendritic cells d3 to be a proliferationrate of dendritic cells due to tumor cells f1 to be the ratecorresponding to the interaction of dendritic cells withCD8+ T cells and g to be the natural death rate of dendriticcells We then have
dD
dt s2 minus f1L + d2N minus d3T( 1113857D minus KDz(M)D minus gD (6)
24 Modeling Cytotoxic CD8+ T Cells Among many factorsthat impact the growth of tumor cells it is well known thatCD8+ T cells are an important component of the immunesystem that kills tumor cells It has been seen that activetumor-specific CD8+ T cells are only present in largenumbers when tumor cells are present [16 34] and aftersome interactions with tumor cells they become inactive[29] It has been observed that mature CD8+ T cells canremove dendritic cells [35 36]
In our model to describe the dynamics of the CD8+
T-cells we will consider f2 to be the rate of interactionbetween dendritic cells and tumor cells to activate CD8+
T cells minus hLT denotes the form of competition betweenCD8+ T cells and tumor cells and minus iL denotes the naturaldeath rate of CD8+ T cells
It has also been seen that CD8+ T cells may be recruitedby the debris from tumor cells lysed by NK cells [37] Toinclude this effect we will incorporate in our model a re-cruitment term that is proportional to the number of cellskilled which is denoted as r1NT We will also need anadditional term that helps describe the regulation andsuppression of CD8+ T-cell activity when there are highlevels of activated CD8+ T cells without responsiveness tocytokines in the system [38 39] )is term is denoted byuNL2 We also include CD8+ T activation by IL-2 immu-notherapy which is in the form of a drug that influences theimmune systemrsquos efficacy and described via a MichaelisndashMenten interaction term [16]
pILI
gI + I (7)
Here I refers to the immunotherapy drug concentrationin the bloodstream )e model for the dynamics for theCD8+ T cell growth population becomes
dL
dt f2DT minus hLT minus uNL
2+ r1NT +
pILI
gI + Iminus KLz(M)L minus iL
(8)
25 Modeling Drug and Vaccine Interventions In this studywe incorporate a variety of external intervention treatmentoptions including tumor-infiltrating lymphocyte (TIL) in-jections as well as chemotherapy and immunotherapy drugsTIL drug intervention may be thought of as an immuno-therapy approach in which the CD8+ T-cells are promotedthrough antigen-specific cytolytic immune cells We do thisby adding the term vL vL(t) in equation (8) and we havedL
dt f2DT minus hLT minus uNL
2+ r1NT +
pILI
gI + Iminus KLz(M)L minus iL + vL
(9)
Computational and Mathematical Methods in Medicine 3
To include the chemotherapy and immunotherapydrugs we describe the dynamics of the respective concen-trations in the blood stream as follows
dMdt
vM(t) minus d4M
dIdt vI(t) minus d5I
(10)
e drug intervention terms in these equations reectthe amount of chemotherapy and immunotherapy druggiven over time Note that we assume that the chemotherapyand immunotherapy drugs will be eliminated from the bodyover time at a rate proportional to its concentration andthese are given by d4M and d5I respectively
26 Overall Model From Sections 21ndash25 we have thefollowing overall model
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T
_N s1 +g1NT
2
h1 + T2 minus c2T minus d1D( )N minus KNz(M)N minus eN
_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD
_L f2DT minus hLT minus uNL2 + r1NT +
pILI
gI + I
minus KLz(M)L minus iL + vL(t)
_M vM(t) minus d4M
_I vI(t) minus d5I(11)
Figure 1 illustrates the network of the dynamics forsystem (11) Sharp arrows represent reproduction or acti-vation and blocked arrows represent inhibition or killinge blocked arrow in red represents the nonlinear in-teraction Note that the suppressing eects of the drug arenot shown explicitly in the gure but included in the model
3 Stability Analysis
In this section we employ mathematical analysis to identifyconditions that can help eliminate tumor cells Also we willdetermine conditions for when tumor-free equilibrium isunstable and the tumor grows without bound
We now consider the system of (11) in the absence oftreatment When we eliminate chemotherapy and immu-notherapy the system reduces to a four-population systemof ODEs Let E(Tlowast Nlowast Dlowast Llowast) be an equilibrium point ofthe system described by the system without drugintervention
At an equilibrium point we havedTdtdNdt
dDdt
dLdt 0 (12)
Since we assume there is constant recruitment throughsource terms s1 and s2 both not equal to zero there is notrivial equilibrium which implies
E Tlowast Nlowast Dlowast Llowast( )ne (0 0 0 0) (13)
For tumor-free equilibrium at an equilibrium point wehave dNdt 0 which yields
s1 + d1DlowastNlowast minus eNlowast 0 (14)
which yields
Nlowast s1
e minus d1Dlowast (15)
Similarly setting dDdt 0 at an equilibrium pointyields
s2 minus d2NlowastDlowast minus gDlowast 0 (16)
Substituting (15) in (16) we get
gd1Dlowast2 minus s2d1 + d2s1 + eg( )Dlowast + es2 0 (17)
Solving the quadratic equation yields
Dlowast12 d1s2 + d2s1 + eg( ) plusmn
d1s2 + d2s1 + eg( )2 minus 4ges2
radic
2gd1
(18)
Hence we have 2 tumor-free equilibrium given byE1(0 Nlowast Dlowast1 0) and E2(0 Nlowast Dlowast2 0)
For these to have biological meaning we need
e minus d1Dlowast gt 0 (19)
d1s2 + d2s1 + egge 2ges2
radic (20)
ese conditions suggest critical values for the death rateand the source term for NK cells to be
e d1Dlowast
s1 2 ges2
radic minus d1s2 + eg( )d2
(21)
in order for the tumor-free equilibrium point to be positivefor biological signicance
NK
CD8+ T
T
Chemotherapy Immunotherapy
D
Figure 1 Network of the dynamics for system (11)
4 Computational and Mathematical Methods in Medicine
)e Jacobian matrix for linearization of system (11)without drug intervention is given by
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(22)
where
a11 a minus 2abTlowast
minus c1Nlowast
minus jDlowast
minus kLlowast
a12 minus c1Tlowast
a13 minus jTlowast
a14 minus kTlowast
a21 2g1N
lowasth1Tlowast
h1 + T( 11138572 minus c2N
lowast
a22 minus c2Tlowast
+ d1Dlowast
minus e
a23 d1Nlowast
a24 0
a31 d3Dlowast
a32 minus d2Dlowast
a33 minus f1Llowast
+ d2Nlowast
minus d3Tlowast
+ g( 1113857
a34 minus f1Dlowast
a41 f2Dlowast
minus hLlowast
+ r1Nlowast
a42 minus uLlowast2
+ r1Tlowast
a43 f2Tlowast
a44 minus hTlowast
+ 2uNlowastLlowast
+ i( 1113857
(23)
Evaluating these terms at the general tumor-free equi-librium point gives
A
a minus c1Nlowast minus jDlowast 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(24)
To solve for the eigenvalues λ we solve the equationdet(A minus λI) 0 or
det
a minus c1Nlowast minus jDlowast minus λ 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e minus λ d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus λ minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i minus λ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
0
(25)which yields
a minus c1Nlowast
minus jDlowast
minus λ( 1113857(minus i minus λ)det(B minus λI) 0 (26)
where matrix B is given by
B d1Dlowast minus e d1N
lowast
minus d2Dlowast minus d2N
lowast + g( 11138571113890 1113891 (27)
It may be noted that trace and determinant for matrix Bcan be computed to be
tr(B) d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 1113857 (28)
det(B) d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 (29)
Solving (26) we compute the eigenvalues to be
λ1 a minus c1Nlowast
minus jDlowast
λ2 minus i(30)
and λ3 and λ4 to be the roots of the equation
λ2 minus λ d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 11138571113858 1113859 + d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 0(31)
From (28) and (29) this can be rewritten as
λ2 minus tr(B)λ + det(B) 0 (32)
Using (19) in equations (28) and (29) we can concludethat
tr(B)lt 0
det(B)gt 0(33)
)is implies that the eigenvalues λ3 and λ4 that are theroots of equation (32) have negative real parts
In order for the tumor-free equilibrium to be stable werequire λ1 lt 0 which implies that if the tumor growth rate ais lesser than the critical value given by c1N
lowast + JDlowast then thetumor population can be eliminated
4 Computational Experiments
In this section we will consider the system of (11) which willbe solved via the RungendashKutta methods )e parametervalues their units and their estimated value are itemizednext which we use in our computations
(i) a tumor growth rate estimated as431lowast10minus 1 dayminus 1 [9]
(ii) b bminus 1 tumor-carrying capacity estimated as217lowast10minus 8 cellsminus 1 [9]
Computational and Mathematical Methods in Medicine 5
(iii) c1 NK cell tumor cell kill rate estimated as35lowast 10minus 6 cellsminus 1 [12]
(iv) c2 NK cell inactivation rate by tumor cells esti-mated as 10lowast 10minus 7 cellsminus 1 dayminus 1 [9]
(v) d1 rate of dendritic cell priming NK cells esti-mated as 10lowast 10minus 6 cellsminus 1 [12]
(vi) d2 NK cell dendritic cell kill rate estimated as40lowast 10minus 6 cellsminus 1 [12]
(vii) d3 rate of tumor cells priming dendritic cellsestimated as 10lowast 10minus 4 Estimate
(viii) e death rate of NK cell estimated as412lowast 10minus 2 dayminus 1 [12]
(ix) f1 CD8+ T cell dendritic cells kill rate estimated
as 10lowast 10minus 8 cellsminus 1 [12](x) f2 rate of dendritic cells priming CD8+ T cell
estimated as 001 cellsminus 1 [12](xi) g death rate of dendritic cells estimated as
24lowast 10minus 2 cellsminus 1 [12](xii) h CD8+ T inactivation rate by tumor cells esti-
mated as 342lowast 10minus 10 cellsminus 1 dayminus 1 [12](xiii) i death rate of CD8+ T cells estimated as
20lowast 10minus 2 dayminus 1 [9](xiv) j dendritic cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xv) k NK cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xvi) s1 source of NK cells estimated as 13lowast 104 cellsminus 1
[12](xvii) s2 source of dendritic cell estimated as
48lowast 102 cellsminus 1 [12](xviii) u regulatory function by Nk cells of CD8+ T cells
estimated as 180lowast 10minus 8 cellminus 2 dayminus 1 [9]
For the first computation we will assume there is noadditional recruitment terms for CD8+ T cells and NK cells(r1 0 g1 0 h1 0) removing some of the regulationsuppression and activation of CD8+ T cells(pI gI u 0) not including the influence of drug killterms (KT KN KD KL 0) no influence of drug andvaccine interventions (vL vM vI 0) along with thecorresponding death rates (d4 d5 0) We will also as-sume d3 0 which corresponds to the new term that hasbeen added to the model to indicate the growth of dendriticcells being impacted by tumor cells We will also assume forsimplicity and illustration purposes of a weak immunesystem that the dynamics start with 100 tumor cells with onenatural killer one dendritic and one CD8+ T cell Figure 2shows the dynamics of each of these cells )e tumor cellsinitially increase to a peak before a full immune clearancestarts
Next we consider the effect of one of the terms insystem (11) corresponding to the dynamics of dendriticcells )is is the term d3TD which includes the influence oftumor growth on the dynamics of dendritic cells that all
the previous studies have not considered Figure 3 illus-trates how not only the dendritic cells are impacted but theCD8+ Tcell dynamics also changes as the proliferation rated3 is doubled For the rest of the simulations we willinclude the effect of d3 and assume the value to be 1 times 10minus 4
Along with d3 we also wanted to study the influence ofthe source term s2 on the dynamics of tumor NK and CD8+
T cells Figure 4 illustrates this behavior When we increasethe source term of dendritic cells it increases NK cells andCD8+ Tcells We also note that these have an effect on tumorgrowth as both NK and CD8+ T cells can lyse tumor cells)is decrease is also seen in this figure )is suggests that anexternal source term of dendritic cells has the potential todecrease tumor growthWe also note that dendritic cells playan important role in recruiting CD8+ T cells earlier in thetumor growth phase
Next to study the effect of TIL drug intervention termonly for the CD8+ T-cell population as an immuno-therapy where the immune cell levels are boosted by theaddition of antigen-specific cytolytic immune cells weincrease the value of vL from 1 to 106 )e result is shownin Figure 5 and we notice that the effect of adding thedrug in small doses does not have a big impact on tumorgrowth
Next we consider the effect of the chemotherapy drugonly that is introduced through the term vM We set vM 1and study the influence of increasing KT in the dynamics oftumor cells Figure 6 illustrates how tumor cells can bereduced through this technique
We want to point out that we also performed the studyon the influence of immunotherapy drug intervention vI
but noticed that even with inclusion of a CD8+ Tactivation described via MichaelisndashMenten interaction givenby
pILI
gI + I (34)
there was negligible effect on the dynamics of all four cellsWe also noted that there was not much effect in the dy-namics of NK cells through the recruitment terms involvingg1 and r1
Next we turn our attention to the effect of the nonlinearterm introduced as an inactivation term which describes theregulation and suppression of CD8+ T-cell activity [38 39]Specifically Figure 7 shows the effect of the nonlinear termin system (11) for parameters u 0 and u 3 times 10minus 10 andthe dynamics show that there is a drastic drop in the numberof cytotoxic CD8+ Tcells while a small increase in tumor cellgrowth
In summary our computations seem to suggest that acombination of immunotherapy through TIL drug in-tervention vM along with chemotherapy through vL providesan optimal way to reduce tumor growth Taking this intoaccount and removing terms that had negligible effects onecan consider the following simplified system that capturesmost prominent features
6 Computational and Mathematical Methods in Medicine
0 50 100Time (days)
0
500
1000
1500
Tum
or ce
lls
(a)
0 50 100Time (days)
0
1
2
3
4
NK
cells
times105
(b)
0 50 100Time (days)
0
500
1000
1500
2000
Den
driti
c cel
ls
(c)
0 50 100Time (days)
0
5
10
15
CD8+ T
cells
times104
(d)
Figure 2 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells
0
500
1000
1500
Tum
or ce
lls
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(a)
0
1
2
3
4
NK
cells
times105
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(b)
Figure 3 Continued
Computational and Mathematical Methods in Medicine 7
0
1
2
3
Den
driti
c cel
ls
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
times104
0 50 100Time (days)
(c)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
0
5
10
15
CD8+ T
cells
times105
0 50 100Time (days)
(d)
Figure 3 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells as the proliferation rate d3 is doubled
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
s2 = 500s2 = 2000s2 = 5000
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
2
4
NK
cells
times105
s2 = 500s2 = 2000s2 = 5000
(b)
0 10 20 30 40 50 60 70 80 90 100Time (days)
s2 = 500s2 = 2000s2 = 5000
times105
0
5
10
15
CD8+ T
cells
(c)
Figure 4 Dynamics of (a) tumor (b) NK and (c) CD8+ T cells as function of source s2
8 Computational and Mathematical Methods in Medicine
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
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Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
To include the chemotherapy and immunotherapydrugs we describe the dynamics of the respective concen-trations in the blood stream as follows
dMdt
vM(t) minus d4M
dIdt vI(t) minus d5I
(10)
e drug intervention terms in these equations reectthe amount of chemotherapy and immunotherapy druggiven over time Note that we assume that the chemotherapyand immunotherapy drugs will be eliminated from the bodyover time at a rate proportional to its concentration andthese are given by d4M and d5I respectively
26 Overall Model From Sections 21ndash25 we have thefollowing overall model
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T
_N s1 +g1NT
2
h1 + T2 minus c2T minus d1D( )N minus KNz(M)N minus eN
_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD
_L f2DT minus hLT minus uNL2 + r1NT +
pILI
gI + I
minus KLz(M)L minus iL + vL(t)
_M vM(t) minus d4M
_I vI(t) minus d5I(11)
Figure 1 illustrates the network of the dynamics forsystem (11) Sharp arrows represent reproduction or acti-vation and blocked arrows represent inhibition or killinge blocked arrow in red represents the nonlinear in-teraction Note that the suppressing eects of the drug arenot shown explicitly in the gure but included in the model
3 Stability Analysis
In this section we employ mathematical analysis to identifyconditions that can help eliminate tumor cells Also we willdetermine conditions for when tumor-free equilibrium isunstable and the tumor grows without bound
We now consider the system of (11) in the absence oftreatment When we eliminate chemotherapy and immu-notherapy the system reduces to a four-population systemof ODEs Let E(Tlowast Nlowast Dlowast Llowast) be an equilibrium point ofthe system described by the system without drugintervention
At an equilibrium point we havedTdtdNdt
dDdt
dLdt 0 (12)
Since we assume there is constant recruitment throughsource terms s1 and s2 both not equal to zero there is notrivial equilibrium which implies
E Tlowast Nlowast Dlowast Llowast( )ne (0 0 0 0) (13)
For tumor-free equilibrium at an equilibrium point wehave dNdt 0 which yields
s1 + d1DlowastNlowast minus eNlowast 0 (14)
which yields
Nlowast s1
e minus d1Dlowast (15)
Similarly setting dDdt 0 at an equilibrium pointyields
s2 minus d2NlowastDlowast minus gDlowast 0 (16)
Substituting (15) in (16) we get
gd1Dlowast2 minus s2d1 + d2s1 + eg( )Dlowast + es2 0 (17)
Solving the quadratic equation yields
Dlowast12 d1s2 + d2s1 + eg( ) plusmn
d1s2 + d2s1 + eg( )2 minus 4ges2
radic
2gd1
(18)
Hence we have 2 tumor-free equilibrium given byE1(0 Nlowast Dlowast1 0) and E2(0 Nlowast Dlowast2 0)
For these to have biological meaning we need
e minus d1Dlowast gt 0 (19)
d1s2 + d2s1 + egge 2ges2
radic (20)
ese conditions suggest critical values for the death rateand the source term for NK cells to be
e d1Dlowast
s1 2 ges2
radic minus d1s2 + eg( )d2
(21)
in order for the tumor-free equilibrium point to be positivefor biological signicance
NK
CD8+ T
T
Chemotherapy Immunotherapy
D
Figure 1 Network of the dynamics for system (11)
4 Computational and Mathematical Methods in Medicine
)e Jacobian matrix for linearization of system (11)without drug intervention is given by
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(22)
where
a11 a minus 2abTlowast
minus c1Nlowast
minus jDlowast
minus kLlowast
a12 minus c1Tlowast
a13 minus jTlowast
a14 minus kTlowast
a21 2g1N
lowasth1Tlowast
h1 + T( 11138572 minus c2N
lowast
a22 minus c2Tlowast
+ d1Dlowast
minus e
a23 d1Nlowast
a24 0
a31 d3Dlowast
a32 minus d2Dlowast
a33 minus f1Llowast
+ d2Nlowast
minus d3Tlowast
+ g( 1113857
a34 minus f1Dlowast
a41 f2Dlowast
minus hLlowast
+ r1Nlowast
a42 minus uLlowast2
+ r1Tlowast
a43 f2Tlowast
a44 minus hTlowast
+ 2uNlowastLlowast
+ i( 1113857
(23)
Evaluating these terms at the general tumor-free equi-librium point gives
A
a minus c1Nlowast minus jDlowast 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(24)
To solve for the eigenvalues λ we solve the equationdet(A minus λI) 0 or
det
a minus c1Nlowast minus jDlowast minus λ 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e minus λ d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus λ minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i minus λ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
0
(25)which yields
a minus c1Nlowast
minus jDlowast
minus λ( 1113857(minus i minus λ)det(B minus λI) 0 (26)
where matrix B is given by
B d1Dlowast minus e d1N
lowast
minus d2Dlowast minus d2N
lowast + g( 11138571113890 1113891 (27)
It may be noted that trace and determinant for matrix Bcan be computed to be
tr(B) d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 1113857 (28)
det(B) d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 (29)
Solving (26) we compute the eigenvalues to be
λ1 a minus c1Nlowast
minus jDlowast
λ2 minus i(30)
and λ3 and λ4 to be the roots of the equation
λ2 minus λ d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 11138571113858 1113859 + d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 0(31)
From (28) and (29) this can be rewritten as
λ2 minus tr(B)λ + det(B) 0 (32)
Using (19) in equations (28) and (29) we can concludethat
tr(B)lt 0
det(B)gt 0(33)
)is implies that the eigenvalues λ3 and λ4 that are theroots of equation (32) have negative real parts
In order for the tumor-free equilibrium to be stable werequire λ1 lt 0 which implies that if the tumor growth rate ais lesser than the critical value given by c1N
lowast + JDlowast then thetumor population can be eliminated
4 Computational Experiments
In this section we will consider the system of (11) which willbe solved via the RungendashKutta methods )e parametervalues their units and their estimated value are itemizednext which we use in our computations
(i) a tumor growth rate estimated as431lowast10minus 1 dayminus 1 [9]
(ii) b bminus 1 tumor-carrying capacity estimated as217lowast10minus 8 cellsminus 1 [9]
Computational and Mathematical Methods in Medicine 5
(iii) c1 NK cell tumor cell kill rate estimated as35lowast 10minus 6 cellsminus 1 [12]
(iv) c2 NK cell inactivation rate by tumor cells esti-mated as 10lowast 10minus 7 cellsminus 1 dayminus 1 [9]
(v) d1 rate of dendritic cell priming NK cells esti-mated as 10lowast 10minus 6 cellsminus 1 [12]
(vi) d2 NK cell dendritic cell kill rate estimated as40lowast 10minus 6 cellsminus 1 [12]
(vii) d3 rate of tumor cells priming dendritic cellsestimated as 10lowast 10minus 4 Estimate
(viii) e death rate of NK cell estimated as412lowast 10minus 2 dayminus 1 [12]
(ix) f1 CD8+ T cell dendritic cells kill rate estimated
as 10lowast 10minus 8 cellsminus 1 [12](x) f2 rate of dendritic cells priming CD8+ T cell
estimated as 001 cellsminus 1 [12](xi) g death rate of dendritic cells estimated as
24lowast 10minus 2 cellsminus 1 [12](xii) h CD8+ T inactivation rate by tumor cells esti-
mated as 342lowast 10minus 10 cellsminus 1 dayminus 1 [12](xiii) i death rate of CD8+ T cells estimated as
20lowast 10minus 2 dayminus 1 [9](xiv) j dendritic cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xv) k NK cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xvi) s1 source of NK cells estimated as 13lowast 104 cellsminus 1
[12](xvii) s2 source of dendritic cell estimated as
48lowast 102 cellsminus 1 [12](xviii) u regulatory function by Nk cells of CD8+ T cells
estimated as 180lowast 10minus 8 cellminus 2 dayminus 1 [9]
For the first computation we will assume there is noadditional recruitment terms for CD8+ T cells and NK cells(r1 0 g1 0 h1 0) removing some of the regulationsuppression and activation of CD8+ T cells(pI gI u 0) not including the influence of drug killterms (KT KN KD KL 0) no influence of drug andvaccine interventions (vL vM vI 0) along with thecorresponding death rates (d4 d5 0) We will also as-sume d3 0 which corresponds to the new term that hasbeen added to the model to indicate the growth of dendriticcells being impacted by tumor cells We will also assume forsimplicity and illustration purposes of a weak immunesystem that the dynamics start with 100 tumor cells with onenatural killer one dendritic and one CD8+ T cell Figure 2shows the dynamics of each of these cells )e tumor cellsinitially increase to a peak before a full immune clearancestarts
Next we consider the effect of one of the terms insystem (11) corresponding to the dynamics of dendriticcells )is is the term d3TD which includes the influence oftumor growth on the dynamics of dendritic cells that all
the previous studies have not considered Figure 3 illus-trates how not only the dendritic cells are impacted but theCD8+ Tcell dynamics also changes as the proliferation rated3 is doubled For the rest of the simulations we willinclude the effect of d3 and assume the value to be 1 times 10minus 4
Along with d3 we also wanted to study the influence ofthe source term s2 on the dynamics of tumor NK and CD8+
T cells Figure 4 illustrates this behavior When we increasethe source term of dendritic cells it increases NK cells andCD8+ Tcells We also note that these have an effect on tumorgrowth as both NK and CD8+ T cells can lyse tumor cells)is decrease is also seen in this figure )is suggests that anexternal source term of dendritic cells has the potential todecrease tumor growthWe also note that dendritic cells playan important role in recruiting CD8+ T cells earlier in thetumor growth phase
Next to study the effect of TIL drug intervention termonly for the CD8+ T-cell population as an immuno-therapy where the immune cell levels are boosted by theaddition of antigen-specific cytolytic immune cells weincrease the value of vL from 1 to 106 )e result is shownin Figure 5 and we notice that the effect of adding thedrug in small doses does not have a big impact on tumorgrowth
Next we consider the effect of the chemotherapy drugonly that is introduced through the term vM We set vM 1and study the influence of increasing KT in the dynamics oftumor cells Figure 6 illustrates how tumor cells can bereduced through this technique
We want to point out that we also performed the studyon the influence of immunotherapy drug intervention vI
but noticed that even with inclusion of a CD8+ Tactivation described via MichaelisndashMenten interaction givenby
pILI
gI + I (34)
there was negligible effect on the dynamics of all four cellsWe also noted that there was not much effect in the dy-namics of NK cells through the recruitment terms involvingg1 and r1
Next we turn our attention to the effect of the nonlinearterm introduced as an inactivation term which describes theregulation and suppression of CD8+ T-cell activity [38 39]Specifically Figure 7 shows the effect of the nonlinear termin system (11) for parameters u 0 and u 3 times 10minus 10 andthe dynamics show that there is a drastic drop in the numberof cytotoxic CD8+ Tcells while a small increase in tumor cellgrowth
In summary our computations seem to suggest that acombination of immunotherapy through TIL drug in-tervention vM along with chemotherapy through vL providesan optimal way to reduce tumor growth Taking this intoaccount and removing terms that had negligible effects onecan consider the following simplified system that capturesmost prominent features
6 Computational and Mathematical Methods in Medicine
0 50 100Time (days)
0
500
1000
1500
Tum
or ce
lls
(a)
0 50 100Time (days)
0
1
2
3
4
NK
cells
times105
(b)
0 50 100Time (days)
0
500
1000
1500
2000
Den
driti
c cel
ls
(c)
0 50 100Time (days)
0
5
10
15
CD8+ T
cells
times104
(d)
Figure 2 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells
0
500
1000
1500
Tum
or ce
lls
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(a)
0
1
2
3
4
NK
cells
times105
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(b)
Figure 3 Continued
Computational and Mathematical Methods in Medicine 7
0
1
2
3
Den
driti
c cel
ls
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
times104
0 50 100Time (days)
(c)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
0
5
10
15
CD8+ T
cells
times105
0 50 100Time (days)
(d)
Figure 3 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells as the proliferation rate d3 is doubled
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
s2 = 500s2 = 2000s2 = 5000
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
2
4
NK
cells
times105
s2 = 500s2 = 2000s2 = 5000
(b)
0 10 20 30 40 50 60 70 80 90 100Time (days)
s2 = 500s2 = 2000s2 = 5000
times105
0
5
10
15
CD8+ T
cells
(c)
Figure 4 Dynamics of (a) tumor (b) NK and (c) CD8+ T cells as function of source s2
8 Computational and Mathematical Methods in Medicine
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
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Submit your manuscripts atwwwhindawicom
)e Jacobian matrix for linearization of system (11)without drug intervention is given by
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(22)
where
a11 a minus 2abTlowast
minus c1Nlowast
minus jDlowast
minus kLlowast
a12 minus c1Tlowast
a13 minus jTlowast
a14 minus kTlowast
a21 2g1N
lowasth1Tlowast
h1 + T( 11138572 minus c2N
lowast
a22 minus c2Tlowast
+ d1Dlowast
minus e
a23 d1Nlowast
a24 0
a31 d3Dlowast
a32 minus d2Dlowast
a33 minus f1Llowast
+ d2Nlowast
minus d3Tlowast
+ g( 1113857
a34 minus f1Dlowast
a41 f2Dlowast
minus hLlowast
+ r1Nlowast
a42 minus uLlowast2
+ r1Tlowast
a43 f2Tlowast
a44 minus hTlowast
+ 2uNlowastLlowast
+ i( 1113857
(23)
Evaluating these terms at the general tumor-free equi-librium point gives
A
a minus c1Nlowast minus jDlowast 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(24)
To solve for the eigenvalues λ we solve the equationdet(A minus λI) 0 or
det
a minus c1Nlowast minus jDlowast minus λ 0 0 0
2g1Nlowasth1Tlowast
h1 + T( 11138572 minus c2N
lowastd1Dlowast minus e minus λ d1N
lowast 0
d3Dlowast minus d2D
lowast minus d2Nlowast + g( 1113857 minus λ minus f1D
lowast
f2Dlowast + r1N
lowast 0 0 minus i minus λ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
0
(25)which yields
a minus c1Nlowast
minus jDlowast
minus λ( 1113857(minus i minus λ)det(B minus λI) 0 (26)
where matrix B is given by
B d1Dlowast minus e d1N
lowast
minus d2Dlowast minus d2N
lowast + g( 11138571113890 1113891 (27)
It may be noted that trace and determinant for matrix Bcan be computed to be
tr(B) d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 1113857 (28)
det(B) d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 (29)
Solving (26) we compute the eigenvalues to be
λ1 a minus c1Nlowast
minus jDlowast
λ2 minus i(30)
and λ3 and λ4 to be the roots of the equation
λ2 minus λ d1Dlowast
minus e( 1113857 minus d2Nlowast
+ g( 11138571113858 1113859 + d1d2NlowastDlowast
minus d1Dlowast
minus e( 1113857 d2Nlowast
+ g( 1113857 0(31)
From (28) and (29) this can be rewritten as
λ2 minus tr(B)λ + det(B) 0 (32)
Using (19) in equations (28) and (29) we can concludethat
tr(B)lt 0
det(B)gt 0(33)
)is implies that the eigenvalues λ3 and λ4 that are theroots of equation (32) have negative real parts
In order for the tumor-free equilibrium to be stable werequire λ1 lt 0 which implies that if the tumor growth rate ais lesser than the critical value given by c1N
lowast + JDlowast then thetumor population can be eliminated
4 Computational Experiments
In this section we will consider the system of (11) which willbe solved via the RungendashKutta methods )e parametervalues their units and their estimated value are itemizednext which we use in our computations
(i) a tumor growth rate estimated as431lowast10minus 1 dayminus 1 [9]
(ii) b bminus 1 tumor-carrying capacity estimated as217lowast10minus 8 cellsminus 1 [9]
Computational and Mathematical Methods in Medicine 5
(iii) c1 NK cell tumor cell kill rate estimated as35lowast 10minus 6 cellsminus 1 [12]
(iv) c2 NK cell inactivation rate by tumor cells esti-mated as 10lowast 10minus 7 cellsminus 1 dayminus 1 [9]
(v) d1 rate of dendritic cell priming NK cells esti-mated as 10lowast 10minus 6 cellsminus 1 [12]
(vi) d2 NK cell dendritic cell kill rate estimated as40lowast 10minus 6 cellsminus 1 [12]
(vii) d3 rate of tumor cells priming dendritic cellsestimated as 10lowast 10minus 4 Estimate
(viii) e death rate of NK cell estimated as412lowast 10minus 2 dayminus 1 [12]
(ix) f1 CD8+ T cell dendritic cells kill rate estimated
as 10lowast 10minus 8 cellsminus 1 [12](x) f2 rate of dendritic cells priming CD8+ T cell
estimated as 001 cellsminus 1 [12](xi) g death rate of dendritic cells estimated as
24lowast 10minus 2 cellsminus 1 [12](xii) h CD8+ T inactivation rate by tumor cells esti-
mated as 342lowast 10minus 10 cellsminus 1 dayminus 1 [12](xiii) i death rate of CD8+ T cells estimated as
20lowast 10minus 2 dayminus 1 [9](xiv) j dendritic cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xv) k NK cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xvi) s1 source of NK cells estimated as 13lowast 104 cellsminus 1
[12](xvii) s2 source of dendritic cell estimated as
48lowast 102 cellsminus 1 [12](xviii) u regulatory function by Nk cells of CD8+ T cells
estimated as 180lowast 10minus 8 cellminus 2 dayminus 1 [9]
For the first computation we will assume there is noadditional recruitment terms for CD8+ T cells and NK cells(r1 0 g1 0 h1 0) removing some of the regulationsuppression and activation of CD8+ T cells(pI gI u 0) not including the influence of drug killterms (KT KN KD KL 0) no influence of drug andvaccine interventions (vL vM vI 0) along with thecorresponding death rates (d4 d5 0) We will also as-sume d3 0 which corresponds to the new term that hasbeen added to the model to indicate the growth of dendriticcells being impacted by tumor cells We will also assume forsimplicity and illustration purposes of a weak immunesystem that the dynamics start with 100 tumor cells with onenatural killer one dendritic and one CD8+ T cell Figure 2shows the dynamics of each of these cells )e tumor cellsinitially increase to a peak before a full immune clearancestarts
Next we consider the effect of one of the terms insystem (11) corresponding to the dynamics of dendriticcells )is is the term d3TD which includes the influence oftumor growth on the dynamics of dendritic cells that all
the previous studies have not considered Figure 3 illus-trates how not only the dendritic cells are impacted but theCD8+ Tcell dynamics also changes as the proliferation rated3 is doubled For the rest of the simulations we willinclude the effect of d3 and assume the value to be 1 times 10minus 4
Along with d3 we also wanted to study the influence ofthe source term s2 on the dynamics of tumor NK and CD8+
T cells Figure 4 illustrates this behavior When we increasethe source term of dendritic cells it increases NK cells andCD8+ Tcells We also note that these have an effect on tumorgrowth as both NK and CD8+ T cells can lyse tumor cells)is decrease is also seen in this figure )is suggests that anexternal source term of dendritic cells has the potential todecrease tumor growthWe also note that dendritic cells playan important role in recruiting CD8+ T cells earlier in thetumor growth phase
Next to study the effect of TIL drug intervention termonly for the CD8+ T-cell population as an immuno-therapy where the immune cell levels are boosted by theaddition of antigen-specific cytolytic immune cells weincrease the value of vL from 1 to 106 )e result is shownin Figure 5 and we notice that the effect of adding thedrug in small doses does not have a big impact on tumorgrowth
Next we consider the effect of the chemotherapy drugonly that is introduced through the term vM We set vM 1and study the influence of increasing KT in the dynamics oftumor cells Figure 6 illustrates how tumor cells can bereduced through this technique
We want to point out that we also performed the studyon the influence of immunotherapy drug intervention vI
but noticed that even with inclusion of a CD8+ Tactivation described via MichaelisndashMenten interaction givenby
pILI
gI + I (34)
there was negligible effect on the dynamics of all four cellsWe also noted that there was not much effect in the dy-namics of NK cells through the recruitment terms involvingg1 and r1
Next we turn our attention to the effect of the nonlinearterm introduced as an inactivation term which describes theregulation and suppression of CD8+ T-cell activity [38 39]Specifically Figure 7 shows the effect of the nonlinear termin system (11) for parameters u 0 and u 3 times 10minus 10 andthe dynamics show that there is a drastic drop in the numberof cytotoxic CD8+ Tcells while a small increase in tumor cellgrowth
In summary our computations seem to suggest that acombination of immunotherapy through TIL drug in-tervention vM along with chemotherapy through vL providesan optimal way to reduce tumor growth Taking this intoaccount and removing terms that had negligible effects onecan consider the following simplified system that capturesmost prominent features
6 Computational and Mathematical Methods in Medicine
0 50 100Time (days)
0
500
1000
1500
Tum
or ce
lls
(a)
0 50 100Time (days)
0
1
2
3
4
NK
cells
times105
(b)
0 50 100Time (days)
0
500
1000
1500
2000
Den
driti
c cel
ls
(c)
0 50 100Time (days)
0
5
10
15
CD8+ T
cells
times104
(d)
Figure 2 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells
0
500
1000
1500
Tum
or ce
lls
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(a)
0
1
2
3
4
NK
cells
times105
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(b)
Figure 3 Continued
Computational and Mathematical Methods in Medicine 7
0
1
2
3
Den
driti
c cel
ls
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
times104
0 50 100Time (days)
(c)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
0
5
10
15
CD8+ T
cells
times105
0 50 100Time (days)
(d)
Figure 3 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells as the proliferation rate d3 is doubled
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
s2 = 500s2 = 2000s2 = 5000
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
2
4
NK
cells
times105
s2 = 500s2 = 2000s2 = 5000
(b)
0 10 20 30 40 50 60 70 80 90 100Time (days)
s2 = 500s2 = 2000s2 = 5000
times105
0
5
10
15
CD8+ T
cells
(c)
Figure 4 Dynamics of (a) tumor (b) NK and (c) CD8+ T cells as function of source s2
8 Computational and Mathematical Methods in Medicine
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
(iii) c1 NK cell tumor cell kill rate estimated as35lowast 10minus 6 cellsminus 1 [12]
(iv) c2 NK cell inactivation rate by tumor cells esti-mated as 10lowast 10minus 7 cellsminus 1 dayminus 1 [9]
(v) d1 rate of dendritic cell priming NK cells esti-mated as 10lowast 10minus 6 cellsminus 1 [12]
(vi) d2 NK cell dendritic cell kill rate estimated as40lowast 10minus 6 cellsminus 1 [12]
(vii) d3 rate of tumor cells priming dendritic cellsestimated as 10lowast 10minus 4 Estimate
(viii) e death rate of NK cell estimated as412lowast 10minus 2 dayminus 1 [12]
(ix) f1 CD8+ T cell dendritic cells kill rate estimated
as 10lowast 10minus 8 cellsminus 1 [12](x) f2 rate of dendritic cells priming CD8+ T cell
estimated as 001 cellsminus 1 [12](xi) g death rate of dendritic cells estimated as
24lowast 10minus 2 cellsminus 1 [12](xii) h CD8+ T inactivation rate by tumor cells esti-
mated as 342lowast 10minus 10 cellsminus 1 dayminus 1 [12](xiii) i death rate of CD8+ T cells estimated as
20lowast 10minus 2 dayminus 1 [9](xiv) j dendritic cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xv) k NK cell tumor cell kill rate estimated as
10lowast 10minus 7 cellsminus 1 [12](xvi) s1 source of NK cells estimated as 13lowast 104 cellsminus 1
[12](xvii) s2 source of dendritic cell estimated as
48lowast 102 cellsminus 1 [12](xviii) u regulatory function by Nk cells of CD8+ T cells
estimated as 180lowast 10minus 8 cellminus 2 dayminus 1 [9]
For the first computation we will assume there is noadditional recruitment terms for CD8+ T cells and NK cells(r1 0 g1 0 h1 0) removing some of the regulationsuppression and activation of CD8+ T cells(pI gI u 0) not including the influence of drug killterms (KT KN KD KL 0) no influence of drug andvaccine interventions (vL vM vI 0) along with thecorresponding death rates (d4 d5 0) We will also as-sume d3 0 which corresponds to the new term that hasbeen added to the model to indicate the growth of dendriticcells being impacted by tumor cells We will also assume forsimplicity and illustration purposes of a weak immunesystem that the dynamics start with 100 tumor cells with onenatural killer one dendritic and one CD8+ T cell Figure 2shows the dynamics of each of these cells )e tumor cellsinitially increase to a peak before a full immune clearancestarts
Next we consider the effect of one of the terms insystem (11) corresponding to the dynamics of dendriticcells )is is the term d3TD which includes the influence oftumor growth on the dynamics of dendritic cells that all
the previous studies have not considered Figure 3 illus-trates how not only the dendritic cells are impacted but theCD8+ Tcell dynamics also changes as the proliferation rated3 is doubled For the rest of the simulations we willinclude the effect of d3 and assume the value to be 1 times 10minus 4
Along with d3 we also wanted to study the influence ofthe source term s2 on the dynamics of tumor NK and CD8+
T cells Figure 4 illustrates this behavior When we increasethe source term of dendritic cells it increases NK cells andCD8+ Tcells We also note that these have an effect on tumorgrowth as both NK and CD8+ T cells can lyse tumor cells)is decrease is also seen in this figure )is suggests that anexternal source term of dendritic cells has the potential todecrease tumor growthWe also note that dendritic cells playan important role in recruiting CD8+ T cells earlier in thetumor growth phase
Next to study the effect of TIL drug intervention termonly for the CD8+ T-cell population as an immuno-therapy where the immune cell levels are boosted by theaddition of antigen-specific cytolytic immune cells weincrease the value of vL from 1 to 106 )e result is shownin Figure 5 and we notice that the effect of adding thedrug in small doses does not have a big impact on tumorgrowth
Next we consider the effect of the chemotherapy drugonly that is introduced through the term vM We set vM 1and study the influence of increasing KT in the dynamics oftumor cells Figure 6 illustrates how tumor cells can bereduced through this technique
We want to point out that we also performed the studyon the influence of immunotherapy drug intervention vI
but noticed that even with inclusion of a CD8+ Tactivation described via MichaelisndashMenten interaction givenby
pILI
gI + I (34)
there was negligible effect on the dynamics of all four cellsWe also noted that there was not much effect in the dy-namics of NK cells through the recruitment terms involvingg1 and r1
Next we turn our attention to the effect of the nonlinearterm introduced as an inactivation term which describes theregulation and suppression of CD8+ T-cell activity [38 39]Specifically Figure 7 shows the effect of the nonlinear termin system (11) for parameters u 0 and u 3 times 10minus 10 andthe dynamics show that there is a drastic drop in the numberof cytotoxic CD8+ Tcells while a small increase in tumor cellgrowth
In summary our computations seem to suggest that acombination of immunotherapy through TIL drug in-tervention vM along with chemotherapy through vL providesan optimal way to reduce tumor growth Taking this intoaccount and removing terms that had negligible effects onecan consider the following simplified system that capturesmost prominent features
6 Computational and Mathematical Methods in Medicine
0 50 100Time (days)
0
500
1000
1500
Tum
or ce
lls
(a)
0 50 100Time (days)
0
1
2
3
4
NK
cells
times105
(b)
0 50 100Time (days)
0
500
1000
1500
2000
Den
driti
c cel
ls
(c)
0 50 100Time (days)
0
5
10
15
CD8+ T
cells
times104
(d)
Figure 2 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells
0
500
1000
1500
Tum
or ce
lls
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(a)
0
1
2
3
4
NK
cells
times105
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(b)
Figure 3 Continued
Computational and Mathematical Methods in Medicine 7
0
1
2
3
Den
driti
c cel
ls
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
times104
0 50 100Time (days)
(c)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
0
5
10
15
CD8+ T
cells
times105
0 50 100Time (days)
(d)
Figure 3 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells as the proliferation rate d3 is doubled
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
s2 = 500s2 = 2000s2 = 5000
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
2
4
NK
cells
times105
s2 = 500s2 = 2000s2 = 5000
(b)
0 10 20 30 40 50 60 70 80 90 100Time (days)
s2 = 500s2 = 2000s2 = 5000
times105
0
5
10
15
CD8+ T
cells
(c)
Figure 4 Dynamics of (a) tumor (b) NK and (c) CD8+ T cells as function of source s2
8 Computational and Mathematical Methods in Medicine
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
0 50 100Time (days)
0
500
1000
1500
Tum
or ce
lls
(a)
0 50 100Time (days)
0
1
2
3
4
NK
cells
times105
(b)
0 50 100Time (days)
0
500
1000
1500
2000
Den
driti
c cel
ls
(c)
0 50 100Time (days)
0
5
10
15
CD8+ T
cells
times104
(d)
Figure 2 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells
0
500
1000
1500
Tum
or ce
lls
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(a)
0
1
2
3
4
NK
cells
times105
0 50 100Time (days)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
(b)
Figure 3 Continued
Computational and Mathematical Methods in Medicine 7
0
1
2
3
Den
driti
c cel
ls
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
times104
0 50 100Time (days)
(c)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
0
5
10
15
CD8+ T
cells
times105
0 50 100Time (days)
(d)
Figure 3 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells as the proliferation rate d3 is doubled
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
s2 = 500s2 = 2000s2 = 5000
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
2
4
NK
cells
times105
s2 = 500s2 = 2000s2 = 5000
(b)
0 10 20 30 40 50 60 70 80 90 100Time (days)
s2 = 500s2 = 2000s2 = 5000
times105
0
5
10
15
CD8+ T
cells
(c)
Figure 4 Dynamics of (a) tumor (b) NK and (c) CD8+ T cells as function of source s2
8 Computational and Mathematical Methods in Medicine
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
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ObesityJournal of
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Computational and Mathematical Methods in Medicine
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Hindawiwwwhindawicom Volume 2018
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Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
0
1
2
3
Den
driti
c cel
ls
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
times104
0 50 100Time (days)
(c)
d3 = 1 times 10ndash4
d3 = 2 times 10ndash4d3 = 4 times 10ndash4
d3 = 8 times 10ndash4
0
5
10
15
CD8+ T
cells
times105
0 50 100Time (days)
(d)
Figure 3 Dynamics of (a) tumor (b) NK (c) dendritic and (d) CD8+ T cells as the proliferation rate d3 is doubled
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
s2 = 500s2 = 2000s2 = 5000
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
2
4
NK
cells
times105
s2 = 500s2 = 2000s2 = 5000
(b)
0 10 20 30 40 50 60 70 80 90 100Time (days)
s2 = 500s2 = 2000s2 = 5000
times105
0
5
10
15
CD8+ T
cells
(c)
Figure 4 Dynamics of (a) tumor (b) NK and (c) CD8+ T cells as function of source s2
8 Computational and Mathematical Methods in Medicine
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
_T aT(1 minus bT) minus c1N + jD + kL( )T minus KTz(M)T_N s1 minus c2T minus d1D( )N minus KNz(M)N minus eN_D s2 minus f1L + d2N minus d3T( )D minus KDz(M)D minus gD_L f2DT minus hLT minus uNL
2 minus KLz(M)L minus iL + vL(t)_M vM(t) minus d4M
(35)
Figure 8 clearly shows that combined chemotherapy andimmunotherapy drug intervention helps reduce the tumorgrowth for system (35) We have used KT 9 times 10minus 2 KD
KN KL 6 times 10minus 2 with vL 106 and vM 1 for ourcomputations
5 Parameter Estimation
In this section we focus on estimating some parameters usedin system (35) based on the measurements of tumor cellsOur goal is to accurately describe the dynamics of tumorgrowth on an individual basis which is very important bothfor growth prediction and designing personalized optimaltherapy schemes (eg when using model predictive control)To demonstrate this let us consider two of the parameters inthe model namely c1 which is the competition rate thatimpacts the dynamics of tumor cells due to natural killercells and d3 which is the parameter-related proliferation ofdendritic cells due to tumor cells Recalling the inuence ofdoubling the latter parameter d3 is illustrated in Figure 3
e purpose of parameter estimation is to identify valuesof parameters for given experimental data In this work wewill demonstrate how to check the reliability of a mathe-matical model to estimate parameters optimally For this weconsider a discrete dataset for tumor dynamics corre-sponding to the values of c1 35 times 10minus 6 and d3 1 times 10minus 4as indicated from the literature (see Figure 9) We thenintroduce randomness in the data by adding some Gaussiannoise to each value in the tumor dynamics We refer to thelatter as the experimental data Tdata (see Figure 10)
Next we make a guess for values of c1 and d3 which arevery dierent from the values used to create the experi-mental data and try to solve the ODE system (35) to obtainthe computed values of the tumor dynamics given byT(c1 d3)
We then set up an error expression E(c1 d3) that is thesum of the squared dierences between the computed valuesT(c1 d3) and the experimental data Tdata as shown inequation (36) Employing an unconstrained nonlinear op-timization algorithm such as the NelderndashMead algorithmthe minimization algorithm searches for a local minimumusing a regression approach is direct search methodattempts to minimize a function of real variables using onlyfunction evaluations without any derivatives If the errorE(c1 d3) is within a user-prescribed tolerance TOL we stopand accept the values of c1 and d3 to be the optimal values Ifnot we use the updated values of parameters c1 and d3 as thenew values and iterate them back to solve the ODE system(35) We continue this until convergence is parameterestimation approach is summarized in Figure 10 and theminimized objective function is given by
E c1 d3( ) sumN
iminus 1T c1 d3( ) minus Tdata( )2 (36)
where E(c1 d3) the least squared error denotes the dif-ferences in the amount of computed tumor cells from thesimulation T(c1 d3) from the observed data Tdata (Figure 9)over N observations
Using same initial conditions with poor guesses for c1 1 times 10minus 8 and d3 1 times 10minus 2 the optimization algorithm es-timates the parameters to be c1 35 times 10minus 6 and
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
vL = 104
vL = 106 vL = 103 vL = 101
vL = 105 vL = 102 vL = 1
Figure 5 Dynamics of tumor cells as the immunotherapy TIL drugintervention term vL is varied
0 20 40 60 80 100Time (days)
0
500
1000
1500
Tum
or ce
lls
KT = 9 times 10ndash3 KN = KD = KL = 6 times 10ndash3
KT = 9 times 10ndash2 KN = KD = KL = 6 times 10ndash2
KT = 9 times 10ndash1 KN = KD = KL = 6 times 10ndash1
KT = 9 times 10ndash4 KN = KD = KL = 6 times 10ndash4
Figure 6 Dynamics of tumor cells as immunotherapy drug in-tervention with constant vM and varying levels of chemotherapydrug kill terms
Computational and Mathematical Methods in Medicine 9
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
500
1000
1500
Tum
or ce
lls
u = 0u = 3 times 10ndash10
(a)
0 10 20 30 40 50 60 70 80 90 100Time (days)
0
05
1
15
2
CD8+ T
cells
times105
u = 0u = 3 times 10ndash10
(b)
Figure 7 Inuence of nonlinearity on dynamics of (a) tumor and (b) CD8+ T cells
0 2 4 6 8 10Time (days)
0
50
100
150
200
250
300
350
400
450
500
Tum
or ce
lls
No drug interventionChemotherapy only
Immunotherapy onlyChemotherapy andimmunotherapy
Figure 8 Inuence of no drug intervention independent drug interventions and combined drug interventions
10 Computational and Mathematical Methods in Medicine
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
d3 045 times 10minus 3 Clearly the algorithm estimates the valuespretty close to the values used originally to create the ex-perimental dataset and the predicted dynamics of the tumorcells for these parameter values along with comparison to theexperimental data is illustrated in Figure 11
6 Discussion and Conclusion
In this work we developed a mathematical model that in-corporated the dynamics of four coupled cell populationsincluding tumor cells natural killer cells dendritic cells andcytotoxic CD8+ T cells that inuence the growth of tumorse novelty in the model was how it combined importantinteractions between growing tumor cells and cells of theinnate and specic immune system coupled with models fordrug delivery to these cell sites A detailed stability analysis ofthe associate ordinary dierential equation system was
performed A variety of computational experiments weresimulated to study the inuence of the dynamics of the fourcell populations on various parameters For example wenoted a signicant eect of the inuence of proliferation rated3 on the dynamics of the cell populations which wasneglected in past studies We noted that the dynamics are afunction of the source terms included in the model eeect of TIL drug intervention as an immunotherapy ap-proach had signicant impact on tumor growth Similarresults were observed for a chemotherapy approach as wellClearly a combined chemotherapy and immunotherapydrug intervention approach was seen to reduce tumorgrowth greatly Another feature of the work is the appli-cation of a parameter estimation algorithm to accuratelypredict proliferation parameters for a given set of data oftumor cell growth Similar algorithms have been used in thepast to characterize properties of soft tissues [40] We werenot only able to capture the data accurately but also were ableto accurately quantify the parameters related to the data
While this paper investigates a system of ordinary dif-ferential equations one must computationally study thecorresponding partial dierential equations (PDES) pre-sented as we developed the model along with uid equationsthat help the drugs to move towards the cancer cells iswill require the use of sophisticated numerical methods likethe nite element methods to solve the associated system ofPDEs is will be the focus of a forthcoming paper
Also we hope to extend our work to apply the modelsand validate them against actual experimental or laboratorydata along with applying machine learning type algorithmsto predict behaviors of the growth of the tumor cells esepredictions can help develop control mechanisms such asdrug therapy is will also be a focus of our future work
Data Availability
All data supporting the results reported have sources pro-vided in the manuscript through references or generatedduring the study
E(c1 d3) gt TOL
T(c1 d3)
E(c1 d3) le TOL
Updatedc1 d3
Guessc1 d3
Stop
Solve ODEsystem (35)
TdataE(c1 d3) = sumN
i=1(T(c1 d3) ndash Tdata)2 Data
Figure 10 Parameter estimation description
700
600
500
400
300
Tum
or ce
lls
200
100
0 20 40Time (days)
60 80 1000
Figure 11 Prediction of the dynamics of tumor cells throughestimated values for c1 35 times 10minus 6 and d3 045 times 10minus 3 in solidline plotted with the data
0 20 40 60 80 100Time (days)
0
100
200
300
400
500
600
700
Tum
or ce
lls
Figure 9 Experimental data for tumor cells for c1 35 times 10minus 6 andd3 1 times 10minus 4 with some noise
Computational and Mathematical Methods in Medicine 11
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
Conflicts of Interest
)e authors declare that they have no conflicts of interest
References
[1] J N Blattman and P D Greenberg ldquoCancer immunotherapya treatment for the massesrdquo Science vol 305 no 5681pp 200ndash205 2004
[2] C R Parish ldquoCancer immunotherapy the past the presentand the futurerdquo Immunology and Cell Biology vol 81 no 2pp 106ndash113 2003
[3] K J OrsquoByrne A G Dalgleish M J Browning W P Stewardand A L Harris ldquo)e relationship between angiogenesis andthe immune response in carcinogenesis and the progression ofmalignant diseaserdquo European Journal of Cancer vol 36 no 2pp 151ndash169 2000
[4] A Ribas and J D Wolchok ldquoCancer immunotherapy usingcheckpoint blockaderdquo Science vol 359 no 6382 pp 1350ndash1355 2018
[5] T H Stewart ldquoImmune mechanisms and tumor dormancyrdquoMedicina (Buenos Aires) vol 56 no 1 pp 74ndash82 1996
[6] B Ribba C Boetsch T Nayak et al ldquoPrediction of theoptimal dosing regimen using a mathematical model of tumoruptake for immunocytokine-based cancer immunotherapyrdquoClinical Cancer Research vol 24 no 14 pp 3325ndash3333 2018
[7] P Veeresha D G Prakasha and H M Baskonus ldquoNewnumerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivativesrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 29 no1 Article ID 013119 2019
[8] L G de Pillis and A Radunskaya ldquoA mathematical model ofimmune response to tumor invasionrdquo in Computational Fluidand Solid Mechanics pp 1661ndash1668 Elsevier ScienceAmsterdam Netherlands 2003
[9] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006
[10] P Bonnotte A Palladini F Pappalardo and S MottaldquoPredictive models in tumor immunologyrdquo Selected TopicsinCancer Modeling E Angelis M A J Chaplain andN Bellomo Eds Springer Science+Business Media LLCBerlin Germany 2008
[11] A Matzavinos M A J Chaplain and V A KuznetsovldquoMathematical modelling of the spatio-temporal response ofcytotoxic T-lymphocytes to a solid tumourrdquo MathematicalMedicine and Biology vol 21 no 1 pp 1ndash34 2004
[12] T Trisilowati S McCue and D Mallet ldquoNumerical solutionof an optimal control model of dendritic cell treatment of agrowing tumourrdquo ANZIAM Journal vol 54 pp 664ndash6802013
[13] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquoBulletin of Mathematical Biology vol 74 no 7 pp 1485ndash15002012
[14] R J de Boer and P Hogeweg ldquoInteractions between mac-rophages and T-lymphocytes tumor sneaking through in-trinsic to helper T cell dynamicsrdquo Journal of 9eoreticalBiology vol 120 no 3 pp 331ndash351 1986
[15] R R Gomis and S Gawrzak ldquoTumor cell dormancyrdquo Mo-lecular Oncology vol 11 no 1 pp 62ndash78 2017
[16] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998
[17] V Kuznetsov and I Makalkin ldquoBifurcation-analysis ofmathematical-model of interactions betweencytotoxic lymphocytes and tumor-cellsmdasheffect of immuno-logical amplification of tumorgrowth and its connection withother phenomena of oncoimmunologyrdquo Biofizika vol 37no 6 pp 1063ndash1070 1992
[18] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994
[19] F Ansarizadeh M Singh and D Richards ldquoModelling oftumor cells regression in response to chemotherapeutictreatmentrdquo Applied Mathematical Modelling vol 48pp 96ndash112 2017
[20] R Eftimie J L Bramson and D J D Earn ldquoInteractionsbetween the immune system and cancer a brief review of non-spatial mathematical modelsrdquo Bulletin of Mathematical Bi-ology vol 73 no 1 pp 2ndash32 2011
[21] S Kato A Goodman V Walavalkar D A BarkauskasA Sharabi and R Kurzrock ldquoHyperprogressors after im-munotherapy analysis of genomic alterations associated withaccelerated growth raterdquo Clinical Cancer Research vol 23no 15 pp 4242ndash4250 2017
[22] F Nani and H I Freedman ldquoAmathematical model of cancertreatment by immunotherapyrdquo Mathematical Biosciencesvol 163 no 2 pp 159ndash199 2000
[23] C J Wheeler D Asha L Gentao J S Yu and K L BlackldquoClinical responsiveness of glioblastoma multiforme to che-motherapy after vaccinationrdquo Clinical Cancer Researchvol 10 no 16 pp 5316ndash5326 2004
[24] National Cancer Institute ldquoUnderstanding chemotherapyrdquoMay 2005 httpwwwncinihgovcancertopicschemotherapy-and-youpage2
[25] M C Perry 9e Chemotherapy Source Book LippincottWilliams ampWilkins Philadelphia PA USA 3rd edition 2001
[26] A Diefenbach E R Jensen A M Jamieson and D H RauletldquoRae1 and H60 ligands of the NKG2D receptor stimulatetumour immunityrdquo Nature vol 413 no 6852 pp 165ndash1712001
[27] Y Kawarda R Ganss N Garbi T Sacher B Arnold andG T Hammerling ldquoNK- and CD8+ T cells-mediate eradi-cation of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotidesrdquo Journal of Immunologyvol 167 no 1 pp 5247ndash5253 2001
[28] N Larmonier J Fraszczak D Lakomy B Bonnotte andE Katsanis ldquoKiller dendritic cells and their potential forcancer immunotherapyrdquo Cancer Immunology Immunother-apy vol 59 no 1 pp 1ndash11 2010
[29] J A Adam and N Bellomo ldquoBasic models of tumor immunesystem interactions-identification analysis and predictionsrdquoin A Survey of Models for Tumor-Immune System DynamicsBirkhauser Basel Switzerland 1997
[30] M Cooper T A Fehniger A Fuchs M Colonna andM A Caligiuri ldquoNK cell and DC interactionsrdquo Trends inImmunology vol 25 no 1 pp 47ndash52 2004
[31] G Ferlazzo M L Tsang L Moretta G MelioliR M Steinman and CMunz ldquoHuman dendritic cells activateresting natural killer (NK) cells and are recognized via theNKp30 receptor by activated NK cellsrdquo 9e Journal of Ex-perimental Medicine vol 195 no 3 pp 343ndash351 2002
12 Computational and Mathematical Methods in Medicine
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
[32] L Fong and E G Engleman ldquoDendritic cells in cancer im-munotherapyrdquo Annual Review of Immunology vol 18 no 1pp 245ndash273 2000
[33] A Moretta ldquoNatural killer cells and dendritic cells rendez-vous in abused tissuesrdquo Nature Reviews Immunology vol 2no 12 pp 957ndash965 2002
[34] I M Tessier J Brostoff and D K Male Immunology MosbySt Louis MO USA 1993
[35] F Castiglione and B Piccoli ldquoOptimal control in a model ofdendritic cell transfection cancer immunotherapyrdquo in Pro-ceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) Maui HI USA December 2004
[36] Y U Wu L Xia M Zhang and X Zhao ldquoImmunodomi-nance analysis through interactions of CD8+ T cells and DCsin lymph nodesrdquo Mathematical Biosciences vol 225 no 1pp 53ndash58 2010
[37] A Huang P Golumbek M Ahmadzadeh E JaffeeD Pardoll and H Levitsky ldquoRole of bone marrow-derivedcells in presenting MHC class I-restricted tumor antigensrdquoScience vol 264 no 5161 pp 961ndash965 1994
[38] S M Gilbertson P D Shah and D A Rowley ldquoNK cellssuppress the generation of Lyt-2+ cytolytic T cells by sup-pressing or eliminating dendritic cellsrdquo Journal of Immu-nology vol 136 no 10 pp 3567ndash3571 1986
[39] A V Gett F Sallusto A Lanzavecchia and J Geginat ldquoTcellfitness determined by signal strengthrdquo Nature Immunologyvol 4 no 4 pp 355ndash360 2003
[40] P Seshaiyer and J D Humphrey ldquoA sub-domain inversefinite element characterization of hyperelastic membranesincluding soft tissuesrdquo Journal of Biomechanical Engineeringvol 125 no 3 pp 363ndash371 2003
Computational and Mathematical Methods in Medicine 13
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom
Stem Cells International
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
MEDIATORSINFLAMMATION
of
EndocrinologyInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Disease Markers
Hindawiwwwhindawicom Volume 2018
BioMed Research International
OncologyJournal of
Hindawiwwwhindawicom Volume 2013
Hindawiwwwhindawicom Volume 2018
Oxidative Medicine and Cellular Longevity
Hindawiwwwhindawicom Volume 2018
PPAR Research
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Immunology ResearchHindawiwwwhindawicom Volume 2018
Journal of
ObesityJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational and Mathematical Methods in Medicine
Hindawiwwwhindawicom Volume 2018
Behavioural Neurology
OphthalmologyJournal of
Hindawiwwwhindawicom Volume 2018
Diabetes ResearchJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Research and TreatmentAIDS
Hindawiwwwhindawicom Volume 2018
Gastroenterology Research and Practice
Hindawiwwwhindawicom Volume 2018
Parkinsonrsquos Disease
Evidence-Based Complementary andAlternative Medicine
Volume 2018Hindawiwwwhindawicom
Submit your manuscripts atwwwhindawicom